$\text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. }$
A) $\frac { 13 \pi } { 12 }$
B) $\frac { 2197 \pi } { 12 }$
C) $\frac { 169 \pi } { 12 }$
D) $\frac { 169 \pi } { 24 }$
E) $\frac { 13 \pi } { 24 }$

Question

$\text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. }$ 
A) $\frac { 13 \pi } { 12 }$
B) $\frac { 2197 \pi } { 12 }$
C) $\frac { 169 \pi } { 12 }$
D) $\frac { 169 \pi } { 24 }$
E) $\frac { 13 \pi } { 24 }$

Quizplus · Accepted Answer

The area of one petal of a polar curve $ r = f(\theta) $ can be found using the formula $ \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta $, where $ \alpha $ and $ \beta $ are the bounds that enclose one petal. For $ r = 13\cos(3\theta) $, the petal repeats every $ \frac{2\pi}{3} $ because the cosine function repeats every $ 2\pi $ and there's a $ 3 $ in $ 3\theta $, dividing the period by $ 3 $. To find the area of one petal, we integrate from $ 0 $ to $ \frac{\pi}{3} $ (since $ 13\cos(3\theta) $ will be positive and form a petal in this interval). The integral is $ \frac{1}{2} \int_{0}^{\frac{\pi}{3}} (13\cos(3\theta))^2 d\theta = \frac{1}{2} \int_{0}^{\frac{\pi}{3}} 169\cos^2(3\theta) d\theta $. Using the double angle formula, $ \cos^2(x) = \frac{1 + \cos(2x)}{2} $, and integrating, we find the area of one petal to be $ \frac{169\pi}{12} $.

Find the area of one petal of r=13cos⁡3θ. \text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. } Find the area of one petal of r=13cos3θ.

$\text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. }$