The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.
$x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }$ about the $y$-axis
Select the correct answer.
A) $V = \frac { 32 } { 3 } \pi$
B) $V = \frac { 8 } { 3 } \pi$
C) $V = \frac { 224 } { 3 } \pi$
D) $V = \frac { 160 } { 3 } \pi$
E) $V = \frac { 64 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The equation represents a circle centered at (0, 2) with radius 2. When rotated about the y-axis, it forms a torus. The volume of the torus is calculated using the formula $ V = 2 \pi^2 R r^2 $, where $ R $ is the distance from the center of the circle to the axis of rotation (2) and $ r $ is the radius of the circle (2). Thus, $ V = 2 \pi^2 \times 2 \times 2^2 = \frac{32}{3} \pi $.

The Region Bounded by the Given Curves Is Rotated About x2+(y−2)2=22x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }x2+(y−2)2=22

The Region Bounded by the Given Curves Is Rotated About $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }$