Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
|
Example 7.2.1
|
|
Working in polar coordinates, calculate the area of the circle .
|
|
|
|
Solution
|
|
|
Mathematical Solution
|
|
•
|
Figure 7.2.1(a) shows that the circle , with , has as its center, and as its radius. Hence, its area is .
|
•
|
The area is computed in polar coordinates via the following definite integral.
|
|
>
|
use plots in
module()
local p1,p2,p3;
p1:=plot(2*cos(t),t=0..Pi,coords=polar):
p2:=plot([[1,0]],style=point,symbol=solidcircle,symbolsize=15,color=green):
p3:=textplot({[.5,.1,typeset(a)],[1.5,.1,typeset(a)]}):
print(display(p1,p2,p3,scaling=constrained,tickmarks=[0,0]));
end module:
end use:
|
|
Figure 7.2.1(a) The circle
|
|
|
|
|
|
|
|
Maple Solution
|
|
•
|
Expression palette: Definite Integral template
Fill in the fields appropriately.
|
•
|
Context Panel: Evaluate and Display Inline
|
|
=
|
|
|
For Maple to provide a graph of the circle , the parameter has to be given a numeric value. If, say, , then the Plot Builder applied to could be used to obtain a graph, provided the coordinate system is set to "polar" via the Options panel.
Graph of via the Plot Builder
|
Enter .
Context Panel: Plots≻Plot Builder
2-D implicit plot
2-D Options, then immediately back to Basic Options
coordinates: polar
axis coordinates: polar
Smart graphing will automatically adjust ranges
|
|
|
|
Alternatively, the following command will graph the circle when . (Select Evaluate in the Context Panel.)
|
|
|
<< Previous Section Section 7.2
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|