Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English

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) Bounded by two functions
Slide Number 3 System Requirement	Here I am using: Ubuntu Linux OS 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. Lower and upper limits are minus 1 and 2.
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.	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.
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.
	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= -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 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 toTra.
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 TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).
	We will type the same values as before and press Enter.
Type g instead of highlighted <Function>.
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.	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 Int.
Point to the menu with various options.	A menu with various options appears
Select Integral( <Function>, <Start x-Value>, <End x-Value>).	Select Integral( <Function>, <Start x-Value>, <End x-Value>).
Type g instead of highlighted <Function>.	Again, we will enter the same values as before.
Press Tab to highlight <Start x-Value>.
Type x(A).
Similarly, type x(B) for End x-Value. 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 and 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 Graphics view, double click on an uppersum rectangle.	In Graphics view, double click on an uppersum rectangle.
In the Redefine text box that opens, click on Object Properties.	In the Redefine text box that opens, click on Object Properties.
Under Color tab, choose yellow. Under Basic tab, uncheck Show Label.	Under Color tab, choose yellow. Under Basic tab, uncheck Show Label.
In the left panel, now click on trapsum. Again, under Basic tab, uncheck Show Label. Click on Color tab. Let us leave the color as the default brown.	In the left panel, now click on trapsum. Again, under Basic tab, uncheck Show Label. Click on Color tab. Let us leave the color as the default brown.
Finally, in the left panel, click on a. Under Basic tab, uncheck Show Label. Click on Color tab and choose blue.	Finally, in the left panel, click on a. Under Color tab, choose blue. Under Basic tab, uncheck Show Label.
Close the Preferences box.	Close the Preferences box.
Click on Options tool and select Rounding , 5 decimal places.	Click on Options tool and select Rounding, 5 decimal places.
Point to text box and to slider n. Drag slider n to 10, then 20, 30, 40 and 50.	Observe the values in the text box as you drag slider n. Drag slider n to 10, then 20, 30, 40 and 50.
Point to all values in Graphics view.	Observe that the upper Reimann and trapezoidal sums remain higher than the integral.
Point to Graphics view.	As n increases, the upper sum decreases. Integral does not change as it is not broken into n shapes. Thus, 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.
	Let us look at the geometrical representation of an indefinite integral.
Slide Number 6 Indefinite Integral: Geometrical Interpretation [[Image:]] Family of integrals of f(x) = 2x. F(x) where y=x2+C, C is any constant. F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x2 + C	Indefinite integral: Geometrical interpretation Parabolas in this figure are members of a family of integrals of f of x, which equals 2x. The family is represented by capital F of x. y is equal to x squared plus capital C, where capital C is any constant. capital F of x is equal to integral of f of x with respect to x. This is equal to integral of 2x with respect to x.
Open a new GeoGebra window.	Let us open a new GeoGebra window
	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. f x in parentheses equals x caret 2 plus 2 space x plus 1
Drag the boundary to see the equation in Algebra view.	Drag the boundary to see the equation in Algebra view.
	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. capital F x in parentheses equals capital I integral f in parentheses
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. h x in parentheses equals capital F prime x in parentheses
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 i x in parentheses equals capital F x in parentheses plus k 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.
The integral function x cubed divided by 3 plus x squared plus x plus 0.7 passes through A.	The integral function x cubed divided by 3 plus x squared plus x plus 0.7 passes through A.
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: The area under a curve 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. Double integral from 0 to 1 and from y2 to y with respect to x then y. This is equal to the double integral from 0 to 1 and from x to squareroot of x. But in the reverse order so that it is first with respect to y then x.
	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 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 equals x and press Enter.
Point to the area between the parabola and the line, from (0,0) to (1,1).	We want to find the area between the parabola and the line from 0 comma 0 to 1 comma 1.
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 IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> ).
Type y for the first function.	Type y for the first function.
Press Tab and type y^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 and type y as the variable.
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.
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.
Select IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> ).
	Now, let us reverse the order of functions and limits.
Type sqrt(x) for the first function and x for the second.	Type sqrt x in parentheses for the first function and x for the second.
Type x as the variable and enter 0 and 1 as start and end values of x.	Type x as the variable and enter 0 and 1 as start and end values of x.
When you press Enter, point to the same output of 1/6.	When you press Enter, you see the same output of 1 divided by 6 as the area.
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 IntegralBetween Function, Function, Start x Value, End x Value. 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) 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 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.

Contributors and Content Editors

Madhurig, Vidhya