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

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Time Narration
00:01 Welcome to this tutorial on Integration using GeoGebra
00:06 In this tutorial, we will use GeoGebra to look at integration to estimate:
00:13 Area Under a Curve (AUC)
00:17 Area bounded by two functions
00:21 Here I am using:

Ubuntu Linux Operating System version 16.04

00:28 GeoGebra 5.0.481.0 hyphen d
00:34 To follow this tutorial, you should be familiar with:

GeoGebra interface, Integration

00:42 For relevant tutorials, please visit our website.
00:47 Definite Integral
00:49 Consider f is a continuous function over interval a, b above the x-axis.
00:57 a and b are called the lower and upper limits of the integral.
01:03 Integral of f of x from a to b with respect to x is the notation for this definite integral.
01:13 It is the area bounded by y equals f of x, x equals a, x equals b and the x-axis.
01:23 Let us calculate the definite integral of this function with respect to x.
01:31 Let us open a new GeoGebra window.
01:35 In the input bar, type the following line and press Enter.
01:41 Note the graph in Graphics view and its equation in Algebra view.
01:48 Using the Slider tool, create a number slider n in Graphics view.
01:56 It should range from 1 to 50 in increments of 1.
02:04 Drag the resulting slider n to 5.
02:09 Under Point, click on Point on Object and click at -1 comma 0 and 2 comma 0 to create A and B.
02:25 Let us look at a few ways to approximate area under the curve.
02:30 These will include upper Riemann and trapezoidal sums as well as integration.
02:37 We will first assign the variable label uppersum to the Upper Riemann Sum in GeoGebra.
02:44 In the input bar, type uppersum is equal to capital U p p.
02:52 The following option appears. Click on it.
02:57 Type g instead of highlighted Function.
03:01 Press Tab to highlight Start x Value.
03:05 Type x A in parentheses.
03:09 Similarly, type x B in parentheses for End x Value and n as Number of Rectangles.

Press Enter.

03:22 Note that five rectangles appear between x equals -1 and 2.
03:30 Under Move Graphics View, click on Zoom In and click in Graphics view.
03:39 Again click on Move Graphics View and drag the background to see all the rectangles properly.
03:48 The upper sum area under the curve AUC adds the area of all these rectangles.
03:57 It is an overestimation of the area under the curve.
04:02 This is because some portion of each rectangle extends above the curve.
04:10 Drag the background to move the graph to the left.
04:15 Let us now assign the variable label trapsum to the Trapezoidal Sum.
04:21 In the input bar, type trapsum is equal to Capital T r a.
04:28 A menu with various options appears.
04:32 Select the following option.
04:36 We will type the same values as before and press Enter.
04:42 In Algebra view, uncheck uppersum to hide it in Graphics view.

Note the shape of the trapezoids.

04:52 Let us now look at the integral as the area under the curve.
04:57 Finally, in the input bar, type capital I n t.
05:04 A menu with various options appears
05:08 Select the following option.
05:12 Again, we will enter the same values as before. And Press Enter.
05:19 In Algebra view, uncheck trapsum to hide it in Graphics view.
05:25 For the integral, the curve is the upper bound of the AUC from x equals -1 to 2.
05:35 In Algebra view, uncheck integral a to hide it in Graphics view.
05:42 Under Slider, click on Text.
05:47 Click in Graphics view to open a text box.
05:51 In the Edit field, type Upper space Sum equals and in Algebra view, click on uppersum.
06:02 Click again in the text box and press Enter.
06:07 Type Trapezoidal space Sum equals and in Algebra view, click on trapsum.
06:16 Click again in the text box and press Enter.
06:21 Type Integral a equals and in Algebra view, click on a.
06:28 In the text box, click OK.
06:31 Click on Move and drag the text box in case you need to see it better.
06:38 Now, click on the text box, click on the Graphics panel and select bold to make the text bold.
06:49 In Algebra view, check a, trapsum and uppersum to show all of them.
06:57 Observe the values in the text box as you drag slider n.
07:03 Trapsum is a better approximation of AUC at high n values.
07:10 Integrating such sums from A to B at high values of n will give us the AUC.
07:18 Let us open a new GeoGebra window
07:22 We will look at the relationship between differentiation and integration.
07:28 Also we will look at finding the integral function through a point A 1 comma 3.
07:35 In the input bar, type the following line and press Enter.
07:41 Let us call integral of f of x capital F of x.
07:47 In the input bar, type the following line and press Enter.
07:53 The integral curve of f of x is red in Graphics view.
08:00 Its equation for capital F of x appears in Algebra view.
08:06 Confirm that this is the integral of f of x.
08:11 Drag the boundary to see the equations properly.
08:16 In the input bar, type the following and press Enter.
08:22 Note that this graph coincides with f of x.
08:28 The equations for f of x and h of x are the same.
08:33 Thus, we can see that integration is the inverse process of differentiation.
08:40 Taking the derivative of an integral, gives back the original function.
08:46 Click on Point tool and create a point at 1 comma 3.
08:54 In the input bar, type the following and press Enter.
09:00 Click on Create Sliders in the window that pops up.
09:05 A slider k appears.
09:08 Double click on slider k.
09:12 Set Min at 0, Max at 5.
09:17 Scroll right to set the Increment to 0.01.
09:24 Close the Preferences box.
09:27 In Algebra view, double-click on i of x and on Object Properties.
09:35 Click on Color tab and select green.
09:41 Close the Preferences box.
09:44 Drag k to make i of x pass through point A.
09:51 Drag the boundary to see i of x properly.
09:56 This function is capital F of x plus 0.7.
10:03 Double Integrals
10:05 Double integrals can be used to find:
10:08 The area under a curve along x and y axes' directions
10:14 The volume under a surface z which is equal to f of x and y
10:21 Double Integral An Example
10:24 Let us find the area between a parabola x equals y squared and the line y equals x.
10:33 The limits are from 0 comma 0 to 1 comma 1.
10:38 This area can be expressed as the double integrals shown here.
10:44 Observe the limits and the order of the integrals in terms of the variables.
10:51 Let us open a new GeoGebra window.

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

11:01 In the input bar, type x equals y caret 2 and press Enter.
11:09 Next, in the input bar, type y equals x and press Enter.
11:16 Click on View tool and select CAS.
11:21 In Algebra view, click top right button to close Algebra view.
11:27 Drag the boundary to make CAS view bigger.
11:32 In CAS view, type Int capital I in line 1.
11:38 A menu with various options appears.
11:42 Scroll down.

Select the following option.

11:47 Type y for the first function.
11:51 Press Tab and type y caret 2 for the second function.
11:57 Press Tab and type y as the variable.
12:02 Press Tab and type 0 and 1 as start and end values of y.
12:10 Press Enter.
12:12 A value 1 divided by 6 appears below the entry.
12:17 This is the area between the parabola and the line from 0 comma 0 to 1 comma 1.
12:25 Let us now express y in terms of x for both functions.
12:31 In CAS view, type Int capital I and choose the same option from the menu as before.
12:42 Now, let us reverse the order of functions and limits.
12:46 Type the following and press Enter.
12:50 You can also use the input bar instead of the CAS view.
12:56 Under View, click on Algebra to see Algebra view again.
13:02 Drag the boundaries to make CAS view smaller.
13:07 In the input bar, type Int capital I.
13:12 From menu, select the following option.
13:16 Type y for the first function.
13:19 Press Tab, type y caret 2 for the second function.
13:25 Press Tab, type 0 as the Start x Value and again press Tab to move to and type 1 as the End x Value.
13:36 Press Enter.
13:38 This will also give you an area a of 0.17 or 1 divided by 6.
13:47 Let us summarize.
13:48 In this tutorial, we have used GeoGebra to understand integration as estimation of:

Area Under a Curve (AUC)

13:59 Area bounded by two functions
14:03 As an assignment:
14:05 Calculate the integrals of f of x and g of x between the limits shown with respect to x.
14:15 Explain the results for g of x.
14:19 As another assignment:

Calculate the shaded areas between these pairs of functions.

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15:06 This is Vidhya Iyer from IIT Bombay, signing off.

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