Find the area of the region enclosed by one loop of the curve.
$r = 7 \cos 8 \theta$
A) $\frac { 49 \pi } { 11 }$
B) $\frac { 49 \pi } { 32 }$
C) $\pi$
D) $49 \pi$
E) $\frac { \pi } { 8 }$

Question

Quizplus · Accepted Answer

To find the area enclosed by one loop of the curve, we use the formula: 
$$A = \frac{1}{2}\int_{	heta_1}^{	heta_2} r^2 \, d	heta$$
where $	heta_1$ and $	heta_2$ are the angles where the curve intersects itself and forms a loop.

In this case, $	heta_1 = 0$ and $	heta_2 = \frac{\pi}{4}$. To see why, we can set $r=0$ and solve for $	heta$ to find the four points where the curve intersects itself. We get $	heta = \frac{k\pi}{8}$ for $k=0,1,3,4$. The angles of the first loop are then $	heta_1 = 0$ and $	heta_2 = \frac{\pi}{4}$.

Substituting $r = 7\cos 8	heta$, we get:
$$A = \frac{1}{2}\int_{0}^{\frac{\pi}{4}} (7\cos 8	heta)^2 \, d	heta = \frac{49\pi}{32}$$

Therefore, the answer is B.

Find the Area of the Region Enclosed by One Loop r=7cos⁡8θr = 7 \cos 8 \thetar=7cos8θ

Find the Area of the Region Enclosed by One Loop $r = 7 \cos 8 \theta$