Evaluate $\int _ { C } x y ^ { 4 } d S$, where $C$ is the right half of the circle $x ^ { 2 } + y ^ { 2 } = 9$.
A) $293.6$
B) $292.6$
C) $291.6$
D) $295.6$
E) $294.6$

Question

Quizplus · Accepted Answer

The integral is evaluated over the right half of the circle $x^2 + y^2 = 9$. Parametrize the curve using $x = 3\cos\theta$, $y = 3\sin\theta$ for $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$. The differential arc length $dS$ is $3d\theta$. The integral becomes $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 3\cos\theta (3\sin\theta)^4 \cdot 3d\theta$, which evaluates to approximately 291.6.

Evaluate ∫Cxy4dS\int _ { C } x y ^ { 4 } d S∫C​xy4dS

Evaluate $\int _ { C } x y ^ { 4 } d S$