View source for Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English

{|border=1
|| '''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'''

[http://www.spoken-tutorial.org/ 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> = x<sup>2</sup> + 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 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. 

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)x<sup>3</sup>+x<sup>2</sup>+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=y<sup>2'''</sup> 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><nowiki>= </nowiki>'''<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=y<sup>2'''</sup>   >>  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) = 0.5x<sup>3</sup>+2x<sup>2</sup>-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-x<sup>2</sup>, y=x'''

[[Image:]]'''x<sup>2</sup>+y<sup>2</sup><nowiki>=9, y=3-x</nowiki>'''

[[Image:]]

'''y=1+x<sup>2</sup>, y=2x<sup>2'''</sup>
|| As another '''assignment''':

Calculate the shaded areas between these pairs of '''functions'''. 
|-
|| '''Slide Number 12'''

'''About Spoken Tutorial project'''
|| The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
|| '''Slide Number 13'''

'''Spoken Tutorial workshops'''
|| The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials

gives certificates on passing online tests.

For more details, please write to us.
|-
|| '''Slide Number 14'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
|| Please post your timed queries on this forum.
|-
|| '''Slide Number 15'''

'''Acknowledgement'''
|| '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
|| 
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}