Difference between revisions of "Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English"

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{|border=1
 
{|border=1
−
| | '''Visual Cue'''
+
|| '''Visual Cue'''
−
| | '''Narration'''
+
|| '''Narration'''
  
 
|-
 
|-
−
| | '''Slide Number 1'''
+
|| '''Slide Number 1'''
  
 
'''Title Slide'''
 
'''Title Slide'''
−
| | Welcome to this '''tutorial''' on '''Integration using GeoGebra'''
+
|| Welcome to this '''tutorial''' on '''Integration using GeoGebra'''
 
|-
 
|-
−
| | '''Slide Number 2'''
+
|| '''Slide Number 2'''
  
 
'''Learning Objectives'''
 
'''Learning Objectives'''
−
| | In this '''tutorial''', we will use '''GeoGebra''' to look at integration to estimate:
+
|| In this '''tutorial''', we will use '''GeoGebra''' to look at integration to estimate:
  
 
'''Area Under a Curve (AUC)'''
 
'''Area Under a Curve (AUC)'''
Line 21: Line 18:
 
Area bounded by two '''functions'''
 
Area bounded by two '''functions'''
 
|-
 
|-
−
| | '''Slide Number 3'''
+
|| '''Slide Number 3'''
  
 
'''System Requirement'''
 
'''System Requirement'''
−
| | Here I am using:
+
|| Here I am using:
  
−
'''Ubuntu Linux''' OS version 16.04
+
'''Ubuntu Linux''' Operating System version 16.04
  
 
'''GeoGebra''' 5.0.481.0-d
 
'''GeoGebra''' 5.0.481.0-d
 
|-
 
|-
−
| | '''Slide Number 4'''
+
|| '''Slide Number 4'''
  
 
'''Pre-requisites'''
 
'''Pre-requisites'''
  
 
[http://www.spoken-tutorial.org/ www.spoken-tutorial.org]
 
[http://www.spoken-tutorial.org/ www.spoken-tutorial.org]
−
| | To follow this '''tutorial''', you should be familiar with:
+
|| To follow this '''tutorial''', you should be familiar with:
  
 
'''GeoGebra''' interface
 
'''GeoGebra''' interface
Line 43: Line 40:
 
For relevant '''tutorials''', please visit our website.
 
For relevant '''tutorials''', please visit our website.
 
|-
 
|-
−
| | '''Slide Number 5'''
+
|| '''Slide Number 5'''
  
 
'''Definite Integral'''
 
'''Definite Integral'''
Line 52: Line 49:
  
 
<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>
 
<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>
−
 
−
 
−
 
  
 
Area bounded by '''y=f(x), x=a, x=b''' and '''x-axis'''
 
Area bounded by '''y=f(x), x=a, x=b''' and '''x-axis'''
−
| |  
+
|| '''Definite Integral'''
−
 
+
−
'''Definite Integral'''
+
−
 
+
  
−
Consider '''f''' is a continuous '''function''' over interval '''a b''' above the '''x-axis'''.  
+
Consider '''f''' is a continuous '''function''' over interval '''a, b''' above the '''x-axis'''.  
  
 
'''a''' and '''b''' are called the lower and upper limits of the integral.  
 
'''a''' and '''b''' are called the lower and upper limits of the integral.  
−
 
  
 
Integral of '''f of x''' from '''a''' to '''b''' with respect to '''x''' is the notation for this definite integral.
 
Integral of '''f of x''' from '''a''' to '''b''' with respect to '''x''' is the notation for this definite integral.
  
  
−
It is the area bounded by '''y''' equals '''f of x, x''' equals '''a, x''' equals '''b''' and the '''x-axis'''.  
+
It is the area bounded by '''y''' equals '''f of x''', '''x''' equals '''a, x''' equals '''b''' and the '''x-axis'''.  
 
|-
 
|-
−
| | '''Slide Number 6'''
+
|| '''Slide Number 6'''
  
 
'''Calculation of a Definite Integral'''
 
'''Calculation of a Definite Integral'''
  
 
Let us calculate the definite integral<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>
 
Let us calculate the definite integral<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>
−
| |  
+
|| Let us calculate the definite integral of this function with respect to '''x'''.
−
 
+
−
Let us calculate the definite integral of this function with respect to '''x'''.
+
 
|-
 
|-
−
| | Open a new '''GeoGebra''' window.  
+
|| Open a new '''GeoGebra''' window.  
−
| | Let us open a new '''GeoGebra''' window.  
+
|| Let us open a new '''GeoGebra''' window.  
 
|-
 
|-
−
| | Type '''g(x)= ‑ 0.5 x^3+ 2 x^2-x+1''' in the '''input bar''' >> '''Enter'''.  
+
|| Type '''g(x)= - 0.5 x^3+ 2 x^2-x+1''' in the '''input bar''' >> '''Enter'''.  
−
| | In the '''input bar''', type the following line and press '''Enter'''.
+
|| In the '''input bar''', type the following line and press '''Enter'''.
 
|-
 
|-
−
| | Point to the graph in '''Graphics''' view and its equation in '''Algebra''' view.  
+
|| Point to the graph in '''Graphics''' view and its equation in '''Algebra''' view.  
−
| | Note the graph in '''Graphics''' view and its equation in '''Algebra''' view.  
+
|| Note the graph in '''Graphics''' view and its equation in '''Algebra''' view.  
 
|-
 
|-
−
| | Click on '''Slider''' tool and click in '''Graphics''' view.
+
|| Click on '''Slider''' tool and click in '''Graphics''' view.  
−
| | Using the '''Slider''' tool, create a number '''slider n''' in '''Graphics''' view.
+
−
 
+
−
It should range from 1 to 50 in increments of 1.
+
−
|-
+
−
| | Leave the '''Number''' radio button checked.
+
−
| |
+
−
|-
+
−
| | Type '''n''' in the '''Name''' field.
+
−
|-
+
−
| | Set 1 as '''Min''', 50 as the '''Max''' and 1 as '''Increment''' >> '''OK'''
+
−
|-
+
−
| | Point to '''slider n''' in '''Graphics''' view.  
+
  
 +
Type '''n''' in the '''Name''' field.
  
 +
Set 1 as '''Min''', 50 as the '''Max''' and 1 as '''Increment''' >> '''OK'''
  
 +
Point to '''slider n''' in '''Graphics''' view.
 +
|| Using the '''Slider''' tool, create a number '''slider n''' in '''Graphics''' view.
 +
It should range from 1 to 50 in increments of 1.
 
|-
 
|-
−
| | Drag '''slider n''' to 5.  
+
|| Drag '''slider n''' to 5.  
−
| | Drag the resulting '''slider n''' to 5.  
+
|| Drag the resulting '''slider n''' to 5.  
 
|-
 
|-
−
| | Click on '''Point on Object''' tool and click at ('''-1,0) '''and '''(2,0) '''to create '''A''' and '''B'''.  
+
|| Click on '''Point on Object''' tool and click at ('''-1,0) '''and '''(2,0) '''to create '''A''' and '''B'''.  
−
| | Under '''Point''', click on '''Point on Object''' and click at ‑1 comma 0 and 2 comma 0 to create '''A''' and '''B'''.  
+
|| Under '''Point''', click on '''Point on Object''' and click at -1 comma 0 and 2 comma 0 to create '''A''' and '''B'''.  
 
|-
 
|-
−
| |  
+
|| Cursor on the GeoGebra interface.
−
| | Let us look at a few ways to approximate '''area under the curve'''.  
+
|| Let us look at a few ways to approximate '''area under the curve'''.  
  
 
These will include '''upper Riemann''' and '''trapezoidal sums''' as well as '''integration'''.  
 
These will include '''upper Riemann''' and '''trapezoidal sums''' as well as '''integration'''.  
Line 120: Line 101:
 
We will first assign the variable label '''uppersum''' to the '''Upper Riemann Sum''' in '''GeoGebra'''.
 
We will first assign the variable label '''uppersum''' to the '''Upper Riemann Sum''' in '''GeoGebra'''.
 
|-
 
|-
−
| | Type '''uppersum=Upp''' in the '''Input Bar'''.
+
|| Type '''uppersum=Upp''' in the '''Input Bar'''.
−
 
+
−
 
+
−
 
+
  
 
Show option.  
 
Show option.  
Line 131: Line 109:
  
 
Click on it.  
 
Click on it.  
−
| | In the '''input bar''', type '''uppersum '''is equal to''' capital U p p'''.  
+
|| In the '''input bar''', type '''uppersum '''is equal to''' capital U p p'''.  
−
 
+
  
 
The following option appears.
 
The following option appears.
−
 
−
 
−
 
−
 
−
 
−
 
 
Click on it.  
 
Click on it.  
 
|-
 
|-
−
| | Type '''g''' instead of highlighted '''<Function>'''.  
+
|| Type '''g''' instead of highlighted '''<Function>'''.  
−
| | Type '''g''' instead of highlighted '''<Function>'''.  
+
|| Type '''g''' instead of highlighted '''<Function>'''.  
 
|-
 
|-
−
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
+
|| Press '''Tab''' to highlight '''<Start x-Value>'''.
−
| | Press '''Tab''' to highlight '''<Start x-Value>'''.  
+
|| Press '''Tab''' to highlight '''<Start x-Value>'''.  
 
|-
 
|-
−
| | Type '''x(A)'''.
+
|| Type '''x(A)'''.
−
| | Type '''x A in parentheses'''.  
+
|| Type '''x A in parentheses'''.  
 
|-
 
|-
−
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles''' >> '''Enter'''
+
|| Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles''' >> '''Enter'''
−
| | Similarly, type '''x B in parentheses''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.  
+
|| Similarly, type '''x B in parentheses''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.  
  
 
Press '''Enter'''.  
 
Press '''Enter'''.  
 
|-
 
|-
−
| | Point to five rectangles between '''x'''<nowiki= -1 and 2. </nowiki>
+
|| Point to five rectangles between '''x'''<nowiki= -1 and 2. </nowiki>
−
| | Note that five rectangles appear between '''x''' equals -1 and 2.  
+
|| Note that five rectangles appear between '''x''' equals -1 and 2.  
 
|-
 
|-
−
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.  
+
|| Under '''Move Graphics View,''' click on '''Zoom In ''' >> click in '''Graphics''' view.  
−
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.  
+
|| Under '''Move Graphics View,''' click on '''Zoom In ''' and click in '''Graphics''' view.  
 
|-
 
|-
−
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.  
+
|| Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.  
−
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.  
+
|| Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.  
 
|-
 
|-
−
| | '''Point''' to '''upper sum area under the curve (AUC).'''  
+
|| '''Point''' to '''upper sum area under the curve (AUC).'''  
−
| | The '''upper sum area under the curve (AUC)''' adds the area of all these rectangles.
+
|| The '''upper sum area under the curve (AUC)''' adds the area of all these rectangles.
 
|-
 
|-
−
| | Point to the rectangles extending above the curve.  
+
|| Point to the rectangles extending above the curve.  
−
| | It is an overestimation of the area under the curve.  
+
|| It is an overestimation of the area under the curve.  
  
 
This is because some portion of each rectangle extends above the curve.  
 
This is because some portion of each rectangle extends above the curve.  
 
|-
 
|-
−
| | Drag the background to move the graph to the left.  
+
|| Drag the background to move the graph to the left.  
−
| | Drag the background to move the graph to the left.  
+
|| Drag the background to move the graph to the left.  
 
|-
 
|-
−
| |  
+
||  
−
| | Let us now assign the variable label '''trapsum''' to the '''Trapezoidal Sum'''.  
+
|| Let us now assign the variable label '''trapsum''' to the '''Trapezoidal Sum'''.  
 
|-
 
|-
−
| | Type '''trapsum=Tra''' in the '''Input bar'''.  
+
|| Type '''trapsum=Tra''' in the '''Input bar'''.  
−
| | In the '''input bar''', type '''trapsum''' is equal to '''Tra'''.
+
|| In the '''input bar''', type '''trapsum''' is equal to '''Tra'''.
 
|-
 
|-
−
| | Point to the menu that appears.  
+
|| Point to the menu that appears.  
−
| | A menu with various options appears.  
+
|| A menu with various options appears.  
 
|-
 
|-
−
| | Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).'''
+
|| Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).'''
−
| | Select the following option.
+
|| Select the following option.
 
|-
 
|-
−
| |  
+
||  
−
| | We will type the same values as before and press '''Enter'''.  
+
|| We will type the same values as before and press '''Enter'''.  
 
|-
 
|-
−
| | Type '''g''' instead of highlighted '''<Function>'''.
+
|| In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.  
−
| |
+
−
|-
+
−
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
+
−
| |
+
−
|-
+
−
| | Type '''x(A)'''.
+
−
| |
+
−
|-
+
−
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.
+
−
| |
+
−
|-
+
−
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.  
+
  
 
Point to trapezoids.
 
Point to trapezoids.
−
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.  
+
|| In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.  
  
 
Note the shape of the trapezoids.  
 
Note the shape of the trapezoids.  
 
|-
 
|-
−
| |  
+
||  
−
| | Let us now look at the integral as the area under the curve.  
+
|| Let us now look at the integral as the area under the curve.  
 
|-
 
|-
−
| | Finally, type '''Int''' in the '''Input Bar'''.  
+
|| Finally, type '''Int''' in the '''Input Bar'''.  
−
| | Finally, in the '''input bar''', type '''Int'''.
+
|| Finally, in the '''input bar''', type '''capital I nt'''.
 
|-
 
|-
−
| | '''Point''' to the menu with various options.
+
|| '''Point''' to the menu with various options.
−
| | A menu with various options appears  
+
|| A menu with various options appears  
 
|-
 
|-
−
| | Select '''Integral( <Function>, <Start x-Value>, <End x-Value>)'''.  
+
|| Select '''Integral( <Function>, <Start x-Value>, <End x-Value>)'''.  
−
| | Select the following option.
+
|| Select the following option.
 
|-
 
|-
−
| | Type '''g''' instead of highlighted '''<Function>'''.
+
|| Enter '''g , x(A), x(B)'''
−
| | Again, we will enter the same values as before.  
+
|| Again, we will enter the same values as before.  
 +
And Press '''Enter.'''
 
|-
 
|-
−
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
+
|| In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.  
−
| |  
+
|| In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
 
|-
 
|-
−
| | Type '''x(A)'''.
+
|| Point to the integrated''' AUC'''.  
−
| |  
+
|| For the integral, the curve is the upper bound of the '''AUC''' from '''x''' equals -1 to 2.
 
|-
 
|-
−
| | Similarly, type '''x(B)''' for '''End x-Value'''.
+
|| In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
−
 
+
|| In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
−
Press '''Enter.'''
+
−
| |  
+
 
|-
 
|-
−
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.  
+
|| Click on '''Text''' tool under '''Slider''' tool.
−
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.  
+
|| Under '''Slider''', click on '''Text'''.
 
|-
 
|-
−
| | Point to the integrated''' AUC'''.  
+
|| Click in '''Graphics''' view to open a '''text box'''.  
−
| | For the integral, the curve is the upper bound of the '''AUC''' from '''x''' equals ‑1 to 2.  
+
|| Click in '''Graphics''' view to open a '''text box'''.  
 
|-
 
|-
−
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
+
|| In the '''Edit''' field, type '''Upper Sum = ''' and in '''Algebra''' view, click on '''uppersum'''.
−
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
+
−
|-
+
−
| | Click on '''Text''' tool under '''Slider''' tool.
+
−
| | Under '''Slider''', click on '''Text'''.
+
−
|-
+
−
| | Click in '''Graphics''' view to open a '''text box'''.
+
−
| | Click in '''Graphics''' view to open a '''text box'''.
+
−
|-
+
−
| | In the '''Edit''' field, type '''Upper Sum = ''' and in '''Algebra''' view, click on '''uppersum'''.
+
  
 
Click again in the '''text box''' and press '''Enter'''.
 
Click again in the '''text box''' and press '''Enter'''.
−
| | In the '''Edit''' field, type '''Upper space Sum equals''' and in '''Algebra''' view, click on '''uppersum'''.
+
|| In the '''Edit''' field, type '''Upper space Sum equals''' and in '''Algebra''' view, click on '''uppersum'''.
  
 
Click again in the '''text box''' and press '''Enter'''.
 
Click again in the '''text box''' and press '''Enter'''.
 
|-
 
|-
−
| | Type '''Trapezoidal Sum =''' and in '''Algebra''' view, click on '''trapsum'''.
+
|| Type '''Trapezoidal Sum =''' and in '''Algebra''' view, click on '''trapsum'''.
  
 
Click again in the '''text box''' and press '''Enter'''.
 
Click again in the '''text box''' and press '''Enter'''.
−
| | Type '''Trapezoidal space Sum equals''' and in '''Algebra''' view, click on '''trapsum'''.
+
|| Type '''Trapezoidal space Sum equals''' and in '''Algebra''' view, click on '''trapsum'''.
  
 
Click again in the '''text box''' and press '''Enter'''.
 
Click again in the '''text box''' and press '''Enter'''.
 
|-
 
|-
−
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.
+
|| Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.
  
 
Click '''OK''' in the '''text box'''.  
 
Click '''OK''' in the '''text box'''.  
−
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.
+
|| Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.
  
 
In the '''text box''', click '''OK'''.  
 
In the '''text box''', click '''OK'''.  
 
|-
 
|-
−
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
+
|| Click on '''Move''' and >> drag the '''text box''' in case you need to see it better.
−
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
+
|| Click on '''Move''' and drag the '''text box''' in case you need to see it better.
 
|-
 
|-
−
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.  
+
|| Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.  
−
| | Now, click on the '''text box'''  and click on the '''Graphics''' panel and select '''bold''' to make the text bold.  
+
|| Now, click on the '''text box''', click on the '''Graphics''' panel and select '''bold''' to make the text bold.  
 
|-
 
|-
−
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.  
+
|| In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.  
−
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.  
+
|| In '''Algebra''' view, check '''a''', '''trapsum''' and '''uppersum''' to show all of them.  
 
|-
 
|-
−
| | Point to text box and to '''slider n'''.  
+
|| Point to text box and to '''slider n'''.  
−
| | Observe the values in the '''text box''' as you drag '''slider n'''.  
+
|| Observe the values in the '''text box''' as you drag '''slider n'''.  
 
|-
 
|-
−
| | Point to '''Graphics''' view.  
+
|| Point to '''Graphics''' view.  
−
| | '''Trapsum''' is a better approximation of '''AUC''' at high '''n''' values.  
+
|| '''Trapsum''' is a better approximation of '''AUC''' at high '''n''' values.  
  
 
'''Integrating''' such '''sums''' from '''A''' to '''B''' at high values of '''n''' will give us the '''AUC'''.  
 
'''Integrating''' such '''sums''' from '''A''' to '''B''' at high values of '''n''' will give us the '''AUC'''.  
Line 293: Line 242:
 
'''F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x<sup>2</sup> + C'''
 
'''F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x<sup>2</sup> + C'''
 
|-
 
|-
−
| | Open a new '''GeoGebra''' window.  
+
|| Open a new '''GeoGebra''' window.  
−
| | Let us open a new '''GeoGebra''' window
+
|| Let us open a new '''GeoGebra''' window
 
|-
 
|-
−
| |  
+
|| Cursor on GeoGebra interface.
−
| | We will look at the relationship between '''differentiation''' and '''integration'''.  
+
|| We will look at the relationship between '''differentiation''' and '''integration'''.  
  
−
Also we will look at finding the '''integral function''' through a point '''A 1 comma 3'''.  
+
Also we will look at finding the '''integral function''' through a point '''A 1 comma 3'''.  
 
|-
 
|-
−
| | Type '''f(x)=x^2+2 x+1''' in the '''Input Bar''' >> '''Enter'''.  
+
|| Type '''f(x)=x^2+2 x+1''' in the '''Input Bar''' >> '''Enter'''.  
−
| | In the '''input bar''', type the following line and press '''Enter'''.  
+
|| In the '''input bar''', type the following line and press '''Enter'''.  
 
|-
 
|-
−
| |  
+
||  
−
| | Let us call '''integral''' of '''f of x capital F of x'''.  
+
|| Let us call '''integral''' of '''f of x capital F of x'''.  
 
|-
 
|-
−
| | Type '''F(x)=Integral(f)''' in the '''Input Bar''' >> '''Enter'''.  
+
|| Type '''F(x)=Integral(f)''' in the '''Input Bar''' >> '''Enter'''.  
−
| | In the '''input bar''', type the following line and press '''Enter'''.  
+
|| In the '''input bar''', type the following line and press '''Enter'''.  
 
|-
 
|-
−
| | Point to the red '''integral''' curve of '''f(x)''' in '''Graphics''' view.  
+
|| Point to the red '''integral''' curve of '''f(x)''' in '''Graphics''' view.  
  
 
Point to equation for '''F(x)=1/3 x<sup>3</sup>+ x<sup>2</sup>+x''' appears in '''Algebra''' view.  
 
Point to equation for '''F(x)=1/3 x<sup>3</sup>+ x<sup>2</sup>+x''' appears in '''Algebra''' view.  
−
| | The '''integral''' curve of '''f of x''' is red in '''Graphics''' view.  
+
|| The '''integral''' curve of '''f of x''' is red in '''Graphics''' view.  
  
 
Its equation for '''capital F of x''' appears in '''Algebra''' view.  
 
Its equation for '''capital F of x''' appears in '''Algebra''' view.  
Line 319: Line 268:
 
Confirm that this is the integral of '''f of x'''.  
 
Confirm that this is the integral of '''f of x'''.  
 
|-
 
|-
−
| | Drag the boundary to see the equations properly.  
+
|| Drag the boundary to see the equations properly.  
−
| | Drag the boundary to see the equations properly.  
+
|| Drag the boundary to see the equations properly.  
 
|-
 
|-
−
| | Type '''h(x)=F'(x)''' in the '''Input Bar''' >> '''Enter'''.  
+
|| Type '''h(x)=F'(x)''' in the '''Input Bar''' >> '''Enter'''.  
−
| | In the '''input bar''', type the following and press '''Enter'''.  
+
|| In the '''input bar''', type the following and press '''Enter'''.  
 
|-
 
|-
−
| | Point to '''F'(x)''' and '''f(x)'''.
+
|| Point to '''F'(x)''' and '''f(x)'''.
−
| | Note that this graph coincides with '''f of x'''.  
+
|| Note that this graph coincides with '''f of x'''.  
  
 
The equations for '''f of x''' and '''h of x''' are the same.
 
The equations for '''f of x''' and '''h of x''' are the same.
Line 334: Line 283:
 
Taking the derivative of an integral, gives back the original '''function'''.  
 
Taking the derivative of an integral, gives back the original '''function'''.  
 
|-
 
|-
−
| | Click on '''Point''' tool and create point '''A''' at '''(1,3)'''.
+
|| Click on '''Point''' tool and create point '''A''' at '''(1,3)'''.
−
| | Click on '''Point''' tool and create a point at '''1 comma 3'''.  
+
|| Click on '''Point''' tool and create a point at '''1 comma 3'''.  
 
|-
 
|-
−
| | Type '''i(x)=F(x)+k''' in the '''Input Bar''' >> '''Enter'''.  
+
|| Type '''i(x)=F(x)+k''' in the '''Input Bar''' >> '''Enter'''.  
−
| | In the '''input bar''', type the following and press '''Enter'''.  
+
|| In the '''input bar''', type the following and press '''Enter'''.  
 
|-
 
|-
−
| | Click on '''Create Sliders''' in the window that pops up.  
+
|| Click on '''Create Sliders''' in the window that pops up.  
−
| | Click on '''Create Sliders''' in the window that pops up.
+
|| Click on '''Create Sliders''' in the window that pops up.
 
|-
 
|-
−
| | Point to '''slider k'''.
+
|| Point to '''slider k'''.
−
| | A '''slider k''' appears.
+
|| A '''slider k''' appears.
 
|-
 
|-
−
| | Double click on '''slider k'''.
+
|| Double click on '''slider k'''.
  
 
Set '''Min''' at 0, '''Max''' at 5 and '''Increment''' to 0.01.  
 
Set '''Min''' at 0, '''Max''' at 5 and '''Increment''' to 0.01.  
  
 
Close the '''Preferences''' window.  
 
Close the '''Preferences''' window.  
−
| | Double click on '''slider k'''.  
+
|| Double click on '''slider k'''.  
  
 
Set '''Min''' at 0, '''Max''' at 5.
 
Set '''Min''' at 0, '''Max''' at 5.
Line 359: Line 308:
 
Close the '''Preferences''' box.  
 
Close the '''Preferences''' box.  
 
|-
 
|-
−
| | Double click on '''i(x)''' in '''Algebra''' view and on '''Object Properties'''.
+
|| Double click on '''i(x)''' in '''Algebra''' view and on '''Object Properties'''.
−
| | In '''Algebra''' view. double-click on '''i of x''' and on '''Object Properties'''.
+
|| In '''Algebra''' view, double-click on '''i of x''' and on '''Object Properties'''.
 
|-
 
|-
−
| | Click on '''Color''' tab and select green.
+
|| Click on '''Color''' tab and select green.
  
 
Close the '''Preferences''' box.  
 
Close the '''Preferences''' box.  
−
| | Click on '''Color''' tab and select green.  
+
|| Click on '''Color''' tab and select green.  
  
 
Close the '''Preferences''' box.  
 
Close the '''Preferences''' box.  
 
|-
 
|-
−
| | Drag '''k''' to make '''i(x)''' pass through point '''A'''.
+
|| Drag '''k''' to make '''i(x)''' pass through point '''A'''.
  
 
Point to integral function '''(1/3)x<sup>3</sup>+x<sup>2</sup>+x+0.7'''.
 
Point to integral function '''(1/3)x<sup>3</sup>+x<sup>2</sup>+x+0.7'''.
−
| | Drag '''k''' to make '''i of x''' pass through point '''A'''.  
+
|| Drag '''k''' to make '''i of x''' pass through point '''A'''.  
 
|-
 
|-
−
| | Drag the boundary to see '''i of x''' properly.  
+
|| Drag the boundary to see '''i of x''' properly.  
−
| | Drag the boundary to see '''i of x''' properly.  
+
|| Drag the boundary to see '''i of x''' properly.  
 
|-
 
|-
−
| | Point to '''F(x)+0.7''': the curve and equation.
+
|| Point to '''F(x)+0.7''': the curve and equation.
−
| | This function is '''capital F of x'''  plus 0.7.  
+
|| This function is '''capital F of x'''  plus 0.7.  
 
|-
 
|-
−
| | '''Slide Number 7'''
+
|| '''Slide Number 7'''
  
 
'''Double Integrals'''
 
'''Double Integrals'''
Line 389: Line 338:
  
 
The volume under a surface '''z=f(x,y)'''
 
The volume under a surface '''z=f(x,y)'''
−
| | '''Double Integrals'''
+
|| '''Double Integrals'''
  
 
'''Double integrals''' can be used to find:
 
'''Double integrals''' can be used to find:
  
−
The '''area under a curve''' along '''x''' and '''y''' '''axes'''’ directions
+
The '''area under a curve''' along '''x''' and '''y''' '''axes'''' directions
  
−
The volume under a surface '''z''' which is equal to '''f of x and y'''
+
The volume under a surface '''z''' which is equal to '''f of x''' and '''y'''
 
|-
 
|-
−
| | '''Slide Number 8'''
+
|| '''Slide Number 8'''
  
 
'''Double Integral-An Example'''
 
'''Double Integral-An Example'''
Line 407: Line 356:
  
 
This area can be expressed as the '''double integral =<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math><nowiki>= </nowiki>'''<math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>
 
This area can be expressed as the '''double integral =<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math><nowiki>= </nowiki>'''<math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>
−
| |  
+
||'''Double Integral-An Example'''
−
 
+
−
'''Double Integral-An Example'''
+
  
 
Let us find the area between a parabola '''x equals y squared''' and the line '''y equals x'''.  
 
Let us find the area between a parabola '''x equals y squared''' and the line '''y equals x'''.  
Line 420: Line 367:
  
 
|-
 
|-
−
| |  
+
||  
−
| | Let us open a new '''GeoGebra''' window.  
+
|| Let us open a new '''GeoGebra''' window.  
  
 
We will first express '''x''' in terms of '''y''', for both '''functions'''.  
 
We will first express '''x''' in terms of '''y''', for both '''functions'''.  
 
|-
 
|-
−
| | In the '''input bar''', type '''x=y<sup>2'''</sup> and press '''Enter'''.  
+
|| In the '''input bar''', type '''x=y<sup>2'''</sup>   >>  press '''Enter'''.  
−
| | In the '''input bar''', type '''x '''equals '''y caret''' 2 and press '''Enter'''.  
+
|| In the '''input bar''', type '''x '''equals '''y caret''' 2 and press '''Enter'''.  
 
|-
 
|-
−
| | Next, in the '''input bar''', type '''y=x''' and press '''Enter'''.  
+
|| Next, in the '''input bar''', type '''y=x''' >> press '''Enter'''.  
−
| | Next, in the '''input bar''', type '''y equals x''' and press '''Enter'''.  
+
|| Next, in the '''input bar''', type '''y equals x''' and press '''Enter'''.  
 
|-
 
|-
−
| | Click on '''View''' tool and select '''CAS'''.  
+
|| Click on '''View''' tool and select '''CAS'''.  
−
| | Click on '''View''' tool and select '''CAS'''.  
+
|| Click on '''View''' tool and select '''CAS'''.  
 
|-
 
|-
−
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.  
+
|| In '''Algebra''' view, click top right button to close '''Algebra''' view.  
−
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.  
+
|| In '''Algebra''' view, click top right button to close '''Algebra''' view.  
 
|-
 
|-
−
| | Drag the boundary to make '''CAS''' view bigger.  
+
|| Drag the boundary to make '''CAS''' view bigger.  
−
| | Drag the boundary to make '''CAS''' view bigger.  
+
|| Drag the boundary to make '''CAS''' view bigger.  
 
|-
 
|-
−
| | In '''CAS''' view, type '''Int''' in line 1.  
+
|| In '''CAS''' view, type '''Int''' in line 1.  
  
 
Point to the menu that appears.  
 
Point to the menu that appears.  
−
| | In '''CAS''' view, type '''Int capital I''' in line 1.  
+
|| In '''CAS''' view, type '''Int capital I''' in line 1.  
  
 
A menu with various options appears.  
 
A menu with various options appears.  
 
|-
 
|-
−
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.  
+
|| Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.  
−
| | Scroll down.  
+
|| Scroll down.  
  
 
Select the following option.
 
Select the following option.
 
|-
 
|-
−
| | Type '''y''' for the first '''function'''.  
+
|| Type '''y''' for the first '''function'''.  
−
| | Type '''y''' for the first '''function'''.  
+
|| Type '''y''' for the first '''function'''.  
 
|-
 
|-
−
| | Press '''Tab''' and type '''y^2''' for the second '''function'''.  
+
|| Press '''Tab''' >> type '''y^2''' for the second '''function'''.  
−
| | Press '''Tab '''and type '''y caret 2''' for the second '''function'''.  
+
|| Press '''Tab '''and type '''y caret 2''' for the second '''function'''.  
 
|-
 
|-
−
| | Press '''Tab''' and type '''y''' as the '''variable'''.
+
|| Press '''Tab''' >> type '''y''' as the '''variable'''.
−
| | Press '''Tab''' and type '''y''' as the '''variable'''.  
+
|| Press '''Tab''' and type '''y''' as the '''variable'''.  
 
|-
 
|-
−
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.  
+
|| Press '''Tab''' >> type 0 and 1 as '''start''' and '''end values''' of '''y'''.  
−
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.  
+
|| Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.  
 
|-
 
|-
−
| | Press '''Enter'''.  
+
|| Press '''Enter'''.  
−
| | Press '''Enter'''.  
+
|| Press '''Enter'''.  
 
|-
 
|-
−
| | Point to the value of 1/6 below the entry.  
+
|| Point to the value of 1/6 below the entry.  
  
 
Point to the area between the parabola and the line from '''(0,0)''' to '''(1,1)'''.  
 
Point to the area between the parabola and the line from '''(0,0)''' to '''(1,1)'''.  
−
| | A value 1 divided by 6 appears below the entry.  
+
|| A value 1 divided by 6 appears below the entry.  
  
 
This is the area between the parabola and the line from '''0 comma 0''' to '''1 comma 1'''.  
 
This is the area between the parabola and the line from '''0 comma 0''' to '''1 comma 1'''.  
 
|-
 
|-
−
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.  
+
|| Let us now express '''y''' in terms of '''x''' for both '''functions'''.  
−
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.  
+
|| Let us now express '''y''' in terms of '''x''' for both '''functions'''.  
 
|-
 
|-
−
| | In '''CAS''' view, type '''Int''' and observe the same menu as before.  
+
|| In '''CAS''' view, type '''Int''' and observe the same menu as before.  
−
| | In '''CAS''' view, type '''Int capital I''' and choose the same option from the menu as before.  
+
|| In '''CAS''' view, type '''Int capital I''' and choose the same option from the menu as before.  
 
|-
 
|-
−
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.  
+
|| Cursor in '''CAS''' view.
−
| |  
+
|| Now, let us reverse the order of '''functions''' and '''limits'''.
 
|-
 
|-
−
| |  
+
|| Type '''sqrt(x)''' for the first function and '''x''' for the second.
−
| | Now, let us reverse the order of '''functions''' and '''limits'''.  
+
|| Type the following and press '''Enter'''.
 
|-
 
|-
−
| | Type '''sqrt(x)''' for the first function and '''x''' for the second.
+
|| Point to the '''input bar'''.
−
| | Type the following and press '''Enter'''.
+
|| You can also use the '''input bar''' instead of the '''CAS''' view.  
 
|-
 
|-
−
| | Point to the '''input bar'''.  
+
|| Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.  
−
| | You can also use the '''input bar''' instead of the '''CAS''' view.  
+
|| Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.  
 
|-
 
|-
−
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.  
+
|| Drag the boundaries to make '''CAS''' view smaller.
−
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.  
+
|| Drag the boundaries to make '''CAS''' view smaller.
 
|-
 
|-
−
| | Drag the boundaries to make '''CAS''' view smaller.
+
|| In the '''input bar''', type '''Int'''.  
−
| | Drag the boundaries to make '''CAS''' view smaller.
+
−
|-
+
−
| |
+
−
In the '''input bar''', type '''Int'''.  
+
  
 
From the menu, select '''IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> )'''.
 
From the menu, select '''IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> )'''.
Line 512: Line 455:
  
 
This will also give you an area a of 0.17 or 1 divided by 6.  
 
This will also give you an area a of 0.17 or 1 divided by 6.  
−
| |  
+
||In the '''input bar''', type '''Int capital I'''.  
−
 
+
−
In the '''input bar''', type '''Int capital I'''.  
+
  
 
From menu, select the following option.
 
From menu, select the following option.
Line 528: Line 469:
 
This will also give you an area '''a''' of 0.17 or 1 divided by 6.  
 
This will also give you an area '''a''' of 0.17 or 1 divided by 6.  
 
|-
 
|-
−
| |  
+
||  
−
| | Let us summarize.
+
|| Let us summarize.
 
|-
 
|-
−
| | '''Slide Number 9'''
+
|| '''Slide Number 9'''
  
 
'''Summary'''
 
'''Summary'''
−
| | In this '''tutorial''', we have used '''GeoGebra''' to understand '''integration''' as estimation of:
+
|| In this '''tutorial''', we have used '''GeoGebra''' to understand '''integration''' as estimation of:
  
 
'''Area Under a Curve''' ('''AUC''')
 
'''Area Under a Curve''' ('''AUC''')
Line 540: Line 481:
 
Area bounded by two '''functions'''
 
Area bounded by two '''functions'''
 
|-
 
|-
−
| | '''Slide Number 10'''
+
|| '''Slide Number 10'''
  
 
'''Assignment'''* Calculate <math>{\int }_{0}^{0.5}f\left(x\right)dx</math>where '''f(x) = 1/(1-x)'''
 
'''Assignment'''* Calculate <math>{\int }_{0}^{0.5}f\left(x\right)dx</math>where '''f(x) = 1/(1-x)'''
Line 547: Line 488:
  
 
'''A, B''' and '''C''' are points where the curve intersects '''x-axis''' (left to right); explain the results
 
'''A, B''' and '''C''' are points where the curve intersects '''x-axis''' (left to right); explain the results
−
| | As an '''assignment''':
+
|| As an '''assignment''':
  
 
Calculate the integrals of '''f of x''' and '''g of x''' between the limits shown with respect to '''x'''.  
 
Calculate the integrals of '''f of x''' and '''g of x''' between the limits shown with respect to '''x'''.  
Line 553: Line 494:
 
Explain the results for '''g of x'''.  
 
Explain the results for '''g of x'''.  
 
|-
 
|-
−
| | '''Slide Number 11'''
+
|| '''Slide Number 11'''
  
 
'''Assignment'''
 
'''Assignment'''
  
 
Calculate the area bounded by the following '''functions''':
 
Calculate the area bounded by the following '''functions''':
−
 
 
[[Image:]]'''y=4x-x<sup>2</sup>, y=x'''
 
[[Image:]]'''y=4x-x<sup>2</sup>, y=x'''
  
Line 566: Line 506:
  
 
'''y=1+x<sup>2</sup>, y=2x<sup>2'''</sup>
 
'''y=1+x<sup>2</sup>, y=2x<sup>2'''</sup>
−
| | As another '''assignment''':
+
|| As another '''assignment''':
  
 
Calculate the shaded areas between these pairs of '''functions'''.  
 
Calculate the shaded areas between these pairs of '''functions'''.  
 
|-
 
|-
−
| | '''Slide Number 12'''
+
|| '''Slide Number 12'''
  
 
'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
−
| | The video at the following link summarizes the '''Spoken Tutorial project'''.
+
|| The video at the following link summarizes the '''Spoken Tutorial project'''.
  
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
−
| | '''Slide Number 13'''
+
|| '''Slide Number 13'''
  
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
−
| | The '''Spoken Tutorial Project '''team:
+
|| The '''Spoken Tutorial Project '''team:
  
 
conducts workshops using spoken tutorials
 
conducts workshops using spoken tutorials
Line 588: Line 528:
 
For more details, please write to us.
 
For more details, please write to us.
 
|-
 
|-
−
| | '''Slide Number 14'''
+
|| '''Slide Number 14'''
  
 
'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
Line 601: Line 541:
  
 
Someone from our team will answer them
 
Someone from our team will answer them
−
| | Please post your timed queries on this forum.
+
|| Please post your timed queries on this forum.
 
|-
 
|-
−
| | '''Slide Number 15'''
+
|| '''Slide Number 15'''
  
 
'''Acknowledgement'''
 
'''Acknowledgement'''
−
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
+
|| '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
  
 
More information on this mission is available at this link.
 
More information on this mission is available at this link.
 
|-
 
|-
−
| |  
+
||  
−
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.
+
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.
  
 
Thank you for joining.
 
Thank you for joining.
 
|-
 
|-
 
|}
 
|}

Revision as of 01:12, 14 January 2019

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Integration using GeoGebra
Slide Number 2

Learning Objectives

In this tutorial, we will use GeoGebra to look at integration to estimate:

Area Under a Curve (AUC)

Area bounded by two functions

Slide Number 3

System Requirement

Here I am using:

Ubuntu Linux Operating System version 16.04

GeoGebra 5.0.481.0-d

Slide Number 4

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with:

GeoGebra interface

Integration

For relevant tutorials, please visit our website.

Slide Number 5

Definite Integral

Consider f is a continuous function over interval [a,b] above x-axis

a is lower limit, b is upper limit

<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>

Area bounded by y=f(x), x=a, x=b and x-axis

Definite Integral

Consider f is a continuous function over interval a, b above the x-axis.

a and b are called the lower and upper limits of the integral.

Integral of f of x from a to b with respect to x is the notation for this definite integral.


It is the area bounded by y equals f of x, x equals a, x equals b and the x-axis.

Slide Number 6

Calculation of a Definite Integral

Let us calculate the definite integral<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>

Let us calculate the definite integral of this function with respect to x.
Open a new GeoGebra window. Let us open a new GeoGebra window.
Type g(x)= - 0.5 x^3+ 2 x^2-x+1 in the input bar >> Enter. In the input bar, type the following line and press Enter.
Point to the graph in Graphics view and its equation in Algebra view. Note the graph in Graphics view and its equation in Algebra view.
Click on Slider tool and click in Graphics view.

Type n in the Name field.

Set 1 as Min, 50 as the Max and 1 as Increment >> OK

Point to slider n in Graphics view.

Using the Slider tool, create a number slider n in Graphics view.

It should range from 1 to 50 in increments of 1.

Drag slider n to 5. Drag the resulting slider n to 5.
Click on Point on Object tool and click at (-1,0) and (2,0) to create A and B. Under Point, click on Point on Object and click at -1 comma 0 and 2 comma 0 to create A and B.
Cursor on the GeoGebra interface. Let us look at a few ways to approximate area under the curve.

These will include upper Riemann and trapezoidal sums as well as integration.

We will first assign the variable label uppersum to the Upper Riemann Sum in GeoGebra.

Type uppersum=Upp in the Input Bar.

Show option.

UpperSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles> )


Click on it.

In the input bar, type uppersum is equal to capital U p p.

The following option appears. Click on it.

Type g instead of highlighted <Function>. Type g instead of highlighted <Function>.
Press Tab to highlight <Start x-Value>. Press Tab to highlight <Start x-Value>.
Type x(A). Type x A in parentheses.
Similarly, type x(B) for End x-Value and n as Number of Rectangles >> Enter Similarly, type x B in parentheses for End x-Value and n as Number of Rectangles.

Press Enter.

Point to five rectangles between x<nowiki= -1 and 2. </nowiki> Note that five rectangles appear between x equals -1 and 2.
Under Move Graphics View, click on Zoom In >> click in Graphics view. Under Move Graphics View, click on Zoom In and click in Graphics view.
Again click on Move Graphics View and drag the background to see all the rectangles properly. Again click on Move Graphics View and drag the background to see all the rectangles properly.
Point to upper sum area under the curve (AUC). The upper sum area under the curve (AUC) adds the area of all these rectangles.
Point to the rectangles extending above the curve. It is an overestimation of the area under the curve.

This is because some portion of each rectangle extends above the curve.

Drag the background to move the graph to the left. Drag the background to move the graph to the left.
Let us now assign the variable label trapsum to the Trapezoidal Sum.
Type trapsum=Tra in the Input bar. In the input bar, type trapsum is equal to Tra.
Point to the menu that appears. A menu with various options appears.
Select TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ). Select the following option.
We will type the same values as before and press Enter.
In Algebra view, uncheck uppersum to hide it in Graphics view.

Point to trapezoids.

In Algebra view, uncheck uppersum to hide it in Graphics view.

Note the shape of the trapezoids.

Let us now look at the integral as the area under the curve.
Finally, type Int in the Input Bar. Finally, in the input bar, type capital I nt.
Point to the menu with various options. A menu with various options appears
Select Integral( <Function>, <Start x-Value>, <End x-Value>). Select the following option.
Enter g , x(A), x(B) Again, we will enter the same values as before.

And Press Enter.

In Algebra view, uncheck trapsum to hide it in Graphics view. In Algebra view, uncheck trapsum to hide it in Graphics view.
Point to the integrated AUC. For the integral, the curve is the upper bound of the AUC from x equals -1 to 2.
In Algebra view, uncheck integral a to hide it in Graphics view. In Algebra view, uncheck integral a to hide it in Graphics view.
Click on Text tool under Slider tool. Under Slider, click on Text.
Click in Graphics view to open a text box. Click in Graphics view to open a text box.
In the Edit field, type Upper Sum = and in Algebra view, click on uppersum.

Click again in the text box and press Enter.

In the Edit field, type Upper space Sum equals and in Algebra view, click on uppersum.

Click again in the text box and press Enter.

Type Trapezoidal Sum = and in Algebra view, click on trapsum.

Click again in the text box and press Enter.

Type Trapezoidal space Sum equals and in Algebra view, click on trapsum.

Click again in the text box and press Enter.

Type Integral a equals and in Algebra view, click on a.

Click OK in the text box.

Type Integral a equals and in Algebra view, click on a.

In the text box, click OK.

Click on Move and >> drag the text box in case you need to see it better. Click on Move and drag the text box in case you need to see it better.
Now, click on the text box and click on the Graphics panel and select bold to make the text bold. Now, click on the text box, click on the Graphics panel and select bold to make the text bold.
In Algebra view, check a, trapsum and uppersum to show all of them. In Algebra view, check a, trapsum and uppersum to show all of them.
Point to text box and to slider n. Observe the values in the text box as you drag slider n.
Point to Graphics view. Trapsum is a better approximation of AUC at high n values.

Integrating such sums from A to B at high values of n will give us the AUC.


F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x2 + C

Open a new GeoGebra window. Let us open a new GeoGebra window
Cursor on GeoGebra interface. We will look at the relationship between differentiation and integration.

Also we will look at finding the integral function through a point A 1 comma 3.

Type f(x)=x^2+2 x+1 in the Input Bar >> Enter. In the input bar, type the following line and press Enter.
Let us call integral of f of x capital F of x.
Type F(x)=Integral(f) in the Input Bar >> Enter. In the input bar, type the following line and press Enter.
Point to the red integral curve of f(x) in Graphics view.

Point to equation for F(x)=1/3 x3+ x2+x appears in Algebra view.

The integral curve of f of x is red in Graphics view.

Its equation for capital F of x appears in Algebra view.

Confirm that this is the integral of f of x.

Drag the boundary to see the equations properly. Drag the boundary to see the equations properly.
Type h(x)=F'(x) in the Input Bar >> Enter. In the input bar, type the following and press Enter.
Point to F'(x) and f(x). Note that this graph coincides with f of x.

The equations for f of x and h of x are the same.

Thus, we can see that integration is the inverse process of differentiation.

Taking the derivative of an integral, gives back the original function.

Click on Point tool and create point A at (1,3). Click on Point tool and create a point at 1 comma 3.
Type i(x)=F(x)+k in the Input Bar >> Enter. In the input bar, type the following and press Enter.
Click on Create Sliders in the window that pops up. Click on Create Sliders in the window that pops up.
Point to slider k. A slider k appears.
Double click on slider k.

Set Min at 0, Max at 5 and Increment to 0.01.

Close the Preferences window.

Double click on slider k.

Set Min at 0, Max at 5.

Scroll right to set the Increment to 0.01.

Close the Preferences box.

Double click on i(x) in Algebra view and on Object Properties. In Algebra view, double-click on i of x and on Object Properties.
Click on Color tab and select green.

Close the Preferences box.

Click on Color tab and select green.

Close the Preferences box.

Drag k to make i(x) pass through point A.

Point to integral function (1/3)x3+x2+x+0.7.

Drag k to make i of x pass through point A.
Drag the boundary to see i of x properly. Drag the boundary to see i of x properly.
Point to F(x)+0.7: the curve and equation. This function is capital F of x plus 0.7.
Slide Number 7

Double Integrals

Double integrals can be used to find:

AUC along x and y axes’ directions

The volume under a surface z=f(x,y)

Double Integrals

Double integrals can be used to find:

The area under a curve along x and y axes' directions

The volume under a surface z which is equal to f of x and y

Slide Number 8

Double Integral-An Example

Let us find the area between parabola x=y2 and the line y=x.

The limits are from (0,0) to (1,1).


This area can be expressed as the double integral =<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math>= <math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>

Double Integral-An Example

Let us find the area between a parabola x equals y squared and the line y equals x.

The limits are from 0 comma 0 to 1 comma 1.

This area can be expressed as the double integrals shown here.

Observe the limits and the order of the integrals in terms of the variables.

Let us open a new GeoGebra window.

We will first express x in terms of y, for both functions.

In the input bar, type x=y2 >> press Enter. In the input bar, type x equals y caret 2 and press Enter.
Next, in the input bar, type y=x >> press Enter. Next, in the input bar, type y equals x and press Enter.
Click on View tool and select CAS. Click on View tool and select CAS.
In Algebra view, click top right button to close Algebra view. In Algebra view, click top right button to close Algebra view.
Drag the boundary to make CAS view bigger. Drag the boundary to make CAS view bigger.
In CAS view, type Int in line 1.

Point to the menu that appears.

In CAS view, type Int capital I in line 1.

A menu with various options appears.

Select IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> ). Scroll down.

Select the following option.

Type y for the first function. Type y for the first function.
Press Tab >> type y^2 for the second function. Press Tab and type y caret 2 for the second function.
Press Tab >> type y as the variable. Press Tab and type y as the variable.
Press Tab >> type 0 and 1 as start and end values of y. Press Tab and type 0 and 1 as start and end values of y.
Press Enter. Press Enter.
Point to the value of 1/6 below the entry.

Point to the area between the parabola and the line from (0,0) to (1,1).

A value 1 divided by 6 appears below the entry.

This is the area between the parabola and the line from 0 comma 0 to 1 comma 1.

Let us now express y in terms of x for both functions. Let us now express y in terms of x for both functions.
In CAS view, type Int and observe the same menu as before. In CAS view, type Int capital I and choose the same option from the menu as before.
Cursor in CAS view. Now, let us reverse the order of functions and limits.
Type sqrt(x) for the first function and x for the second. Type the following and press Enter.
Point to the input bar. You can also use the input bar instead of the CAS view.
Under View, click on Algebra to see Algebra view again. Under View, click on Algebra to see Algebra view again.
Drag the boundaries to make CAS view smaller. Drag the boundaries to make CAS view smaller.
In the input bar, type Int.

From the menu, select IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> ).

Type y for the first function.

Press Tab, type y caret 2 for the second function.

Press Tab, type 0 as the Start Value and again press Tab to move to and type 1 as the End Value.

Press Enter.

This will also give you an area a of 0.17 or 1 divided by 6.

In the input bar, type Int capital I.

From menu, select the following option.

Type y for the first function.

Press Tab, type y caret 2 for the second function.

Press Tab, type 0 as the Start x Value and again press Tab to move to and type 1 as the End x Value.

Press Enter.

This will also give you an area a of 0.17 or 1 divided by 6.

Let us summarize.
Slide Number 9

Summary

In this tutorial, we have used GeoGebra to understand integration as estimation of:

Area Under a Curve (AUC)

Area bounded by two functions

Slide Number 10

Assignment* Calculate <math>{\int }_{0}^{0.5}f\left(x\right)dx</math>where f(x) = 1/(1-x) Calculate <math>{\int }_{x\left(A\right)}^{x\left(B\right)}g\left(x\right)dx</math>and <math>{\int }_{x\left(B\right)}^{x\left(C\right)}g\left(x\right)dx</math>where g(x) = 0.5x3+2x2-x-3.75


A, B and C are points where the curve intersects x-axis (left to right); explain the results

As an assignment:

Calculate the integrals of f of x and g of x between the limits shown with respect to x.

Explain the results for g of x.

Slide Number 11

Assignment

Calculate the area bounded by the following functions: [[Image:]]y=4x-x2, y=x

[[Image:]]x2+y2=9, y=3-x

[[Image:]]

y=1+x2, y=2x2

As another assignment:

Calculate the shaded areas between these pairs of functions.

Slide Number 12

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Slide Number 13

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Slide Number 14

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Slide Number 15

Acknowledgement

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This is Vidhya Iyer from IIT Bombay, signing off.

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Contributors and Content Editors

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