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

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

Latest revision as of 05:35, 15 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.
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 Capital T ra.
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 >> 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.

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.
Point to 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. 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: y=4x-x2, y=x

x2+y2=9, y=3-x


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

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Slide Number 14

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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