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 $\int _ { C } x y ^ { 4 } d S$ over the right half of the circle $x ^ { 2 } + y ^ { 2 } = 9$ can be evaluated using polar coordinates. The circle has radius 3, and the right half corresponds to $\theta$ from $-\pi/2$ to $\pi/2$. The integral becomes $\int_{-\pi/2}^{\pi/2} (3\cos\theta)(3^4\sin^4\theta) \cdot 3 \, d\theta$, which simplifies to $3^6 \int_{-\pi/2}^{\pi/2} \cos\theta \sin^4\theta \, d\theta$. Solving this integral gives approximately 291.6.

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

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