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Aim:
\[1.\ \text{Graph the functions}\ \hspace{22cm}\\ \hspace{1cm} \text{c (constant), x^n, n ∈ R, sin x, cos x},\ sec^2\ x,\ cosec^2\ x,\ \hspace{15cm}\\ \text{secx tanx and cosecxcotx. Find their indefinite integrals}\ \hspace{10cm}\]
\[2.\ \text{Evaluate the definite integral}\ \int_{a}^{b} x^2\ dx\ \hspace{15cm}\\ \text{and relate it to the area under the curve y=f(x)}\ \hspace{13cm}\\ \text{between x – axis, x = a and x = b}\ \hspace{10cm}\]
\[3.\ \text{Find the volume of the solid generated by the revolution}\ \hspace{15cm}\\ \text{of the area bounded by y=f(x)}\ \hspace{10cm}\\ \text{x – axis, x = a and x = b about x axis}\ \hspace{10cm}\]
Procedure:
1. Indefinite Integrals.
\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{8cm}\]
\[\color{green}{Step\ 2:}\ \text{To draw the graph of the function}\ x^2\ by\ using\ input\ bar\ \hspace{10cm}\\ \text{to type x^2 and press the Enter key}\]
\[\color{green}{Step\ 3:}\ \text{To evaluate the indefinite integral of the given function f(x)}\hspace{13cm}\\ \text{by using input command type integral f(x) and press the Enter key}\ \hspace{10cm}\]
\[\color{green}{Step\ 4:}\ \text{create an input box for the function f(x)}\ \hspace{15cm}\\ \text{and link with the given function f(x)}\ \hspace{10cm}\]
\[\color{green}{Step\ 5:}\ \text{use input box enters the given functions}\ \hspace{15cm}\\ \text{and observe the graph of indefinite integrals}\ \hspace{10cm}\]
Output:
output for indefinite integrals
| Function f(x) | |
| 5 | 5x |
| x2 | 1/3 x3 |
| ex | ex |
| sin x | – cos x |
| cos x | sin x |
| sec2x | tan x |
2. Definite Integrals.
\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 window}\ \hspace{15cm}\]
\[\color{green}{Step\ 2:}\ \text{To draw the graph of the function}\ x^2\ by\ using\ input\ bar\ \hspace{10cm}\\ \text{to type x^2 and press the Enter key}\]
\[\color{green}{Step\ 3:}\ \text{Create two number sliders a and b.}\ \hspace{15cm}\]
\[\color{green}{Step\ 4:}\ \text{Create two input boxes link with slider a and b.}\ \hspace{15cm}\]
\[\color{green}{Step\ 5:}\ \text{Create an input box for the given function link with f(x)}\ \ \hspace{15cm}\]
\[\color{green}{Step\ 6:}\ \text{Evaluate the definite integral}\ \int_{0}^{3}\ x^3\ dx\ \hspace{15cm}\\ \text{and relate it to the area under the curve y =}\ x^2\ \text{between x – axis,}\ \hspace{10cm}\\ \text{x=0 and x=3. By using the input command A=Integral(f,0,3).}\ \hspace{10cm}\]
\[\color{green}{Step\ 7:}\ \text{Drag or move the sliders a and b,}\ \hspace{15cm}\\ \text{observe the area under curve}\ \hspace{12cm}\]
\[\color{green}{Step\ 8:}\ \text{Change the values of f(x), a and b in the}\ \hspace{15cm}\\ \text{respective input boxes and record the}\ \hspace{10cm}\\ \text{corresponding values of areas.}\ \hspace{10cm}\]
Output:
output for definite integrals and area under the curve
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| Function f(x) | x = a | x = b | Area between the f(x), y = 0, x = a and x = b | |
| x2 | 0 | 3 | 9 | |
| x | 0 | 6 | 18 | |
| x3 | 1 | 2 | 3.75 |
3. Volume of the Solid.
\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 window and 3D graphics window}\ \hspace{14cm}\]
\[\color{green}{Step\ 2:}\ \text{To draw the graph of the given function x}\ \hspace{15cm}\\ \text{in the interval x = 0 to x = 3 by using the}\ \hspace{10cm}\\ \text{input bar to type if(0<=x<=3, x)}\ \hspace{10cm}\]
\[\color{green}{Step\ 3:}\ \text{Create an angle slider for α.}\ \hspace{15cm}\]
\[\color{green}{Step\ 4:}\ \text{To generate the surface of the solid formed by the}\ \hspace{15cm}\\ \text{revolution of the area bounded by y =x, x-axis, x = 0 and x = 3}\ \hspace{10cm}\\ \text{about x – axis by using the command Surface{f,α, xAxis}}\ \hspace{10cm}\]
\[\color{green}{Step\ 5:}\ \text{Right click on the slider, choose Animation on.}\ \ \hspace{15cm}\]
\[\color{green}{Step\ 6:}\ \text{To find the volume of the solid by}\ \hspace{15cm}\\ \text{using the input command V = pi Integral(f^2,0,3)}\ \hspace{10cm}\\ \text{or V= π Integral(f^2,0,3)}\ \hspace{10cm}\]
\[\color{green}{Step\ 7:}\ \text{Rotate the line y = x and the surface}\ \hspace{15cm}\\ \text{revolution of the area bounded by y =x, x-axis, x = 0 and x = 3}\ \hspace{10cm}\\ \text{about x – axis by using the command Surface{f,α, xAxis}}\ \hspace{10cm}\]
Output:
output for the volume of a solid revolution of surface generated by a curve
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| Function f(x) | x = a | x = b | |
| x | 0 | 3 | 28.27 |
| x2 | 0 | 3 | 152.68 |
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