Find the area of the specified region.
-Inside the limacon $r = 7 + 6 \sin \theta$
A) $67 \pi$
B) $134 \pi - 84$
C) $134 \pi$
D) $67 \pi + 84$

Question

Quizplus · Accepted Answer

The area inside a limacon described by $r = a + b\sin\theta$ (where $a = 7$ and $b = 6$ in this case) can be found using the formula for the area of a polar curve, $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$. For a complete limacon, $\alpha = 0$ and $\beta = 2\pi$, giving $A = \frac{1}{2} \int_{0}^{2\pi} (7 + 6\sin\theta)^2 d\theta$. This simplifies to $A = \frac{1}{2} \int_{0}^{2\pi} (49 + 84\sin\theta + 36\sin^2\theta) d\theta$. The integral of $\sin\theta$ over a full period is 0, and the integral of $\sin^2\theta$ over a full period is $\pi$. Thus, the area is $A = \frac{1}{2} [49(2\pi) + 0 + 36\pi] = 67\pi$.

Find the Area of the Specified Region r=7+6sin⁡θr = 7 + 6 \sin \thetar=7+6sinθ

Find the Area of the Specified Region $r = 7 + 6 \sin \theta$