Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English-timed
From Script | Spoken-Tutorial
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.
Thank you for joining. |