Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Working in polar coordinates, calculate the area of the circle r=2 a cosθ.
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
Figure 7.2.1(a) The circle r=2 a cosθ
Expression palette: Definite Integral template
Fill in the fields appropriately.
Context Panel: Evaluate and Display Inline
12∫0π2 a cosθ2 ⅆθ = π⁢a2
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
axis coordinates: polar
Smart graphing will automatically adjust ranges
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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