PoojaMoolya: Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this '''tutorial''' on '''Integration using GeoGebra''' |- || 00:06 || In this '''tutorial''', we will u..."

2020-10-21T07:14:02Z

Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this '''tutorial''' on '''Integration using GeoGebra''' |- || 00:06 || In this '''tutorial''', we will u..."

New page

{|border=1
|| '''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'''.
|-
|| 14:28
|| The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
|| 14:37
|| The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials and gives certificates on passing online tests.

|-
||14:46
|| For more details, please write to us.
|-
|| 14:49
|| Please post your timed queries on this forum.
|-
|| 14:53
|| '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
|| 15:06
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

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

PoojaMoolya: Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this '''tutorial''' on '''Integration using GeoGebra''' |- || 00:06 || In this '''tutorial''', we will u..."