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 r=2 a cosθ.
Solution
Mathematical Solution
Figure 7.2.1(a) shows that the circle r=2 a cosθ, with θ∈0,π, has x,y=a,0 as its center, and a as its radius. Hence, its area is π a2.
The area is computed in polar coordinates via the following definite integral.
A=12∫0π2 a cosθ2 ⅆθ=π a2
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 r=2 a cosθ
Maple Solution
Expression palette: Definite Integral template Fill in the fields appropriately.
Context Panel: Evaluate and Display Inline
12∫0π2 a cosθ2 ⅆθ = πa2
For Maple to provide a graph of the circle r=2 a cosθ, the parameter a has to be given a numeric value. If, say, a=1, then the Plot Builder applied to r=2 cosθ could be used to obtain a graph, provided the coordinate system is set to "polar" via the Options panel.
Graph of r=2 cosθ via the Plot Builder
Enter r=2 cosθ.
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
r=2 cosθ→
Alternatively, the following command will graph the circle when a=1. (Select Evaluate in the Context Panel.)
plot2 cosθ,θ=0..π,coords=polar, scaling=constrained
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