Textbook Question
In Exercises 61–63, test for symmetry with respect to
a. the polar axis.
b. the line θ = π/2.
c. the pole.
r = 5 + 3 cos θ
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9. Polar Equations
Polar Coordinate System
Multiple Choice
The graph of the polar curve is shown above for . What is the area of the shaded region?
A
B
C
D
Verified step by step guidance1
Identify the given polar curve: \(r = \pi\), which is a circle with radius \(\pi\) centered at the origin in polar coordinates.
Recall the formula for the area enclosed by a polar curve from \(\theta = a\) to \(\theta = b\): \(A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta\).
Substitute the given values into the formula: since \(r = \pi\) is constant, the integral becomes \(A = \frac{1}{2} \int_{0}^{3\pi} (\pi)^2 \, d\theta\).
Simplify the integral by factoring out constants: \(A = \frac{1}{2} \pi^2 \int_{0}^{3\pi} d\theta\).
Evaluate the integral \(\int_{0}^{3\pi} d\theta = 3\pi\), then multiply by the constants to express the area as \(A = \frac{1}{2} \pi^2 \times 3\pi\).
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