Using cylindrical coordinates, the volume of the solid enclosed by $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 , x \geq 0$ and $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$ is

A) $\frac { 16 ( 3 \pi - 4 ) } { 9 }$
B) $\frac { 16 ( 3 \pi - 4 ) } { 7 }$
C) $\frac { 16 ( 3 \pi - 4 ) } { 5 }$
D) $\frac { 16 ( 3 \pi - 4 ) } { 3 }$
E) $\frac { 16 ( 3 \pi - 4 ) } { 11 }$

Question

Using cylindrical coordinates, the volume of the solid enclosed by $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 , x \geq 0$ and $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$ is

A) $\frac { 16 ( 3 \pi - 4 ) } { 9 }$
B) $\frac { 16 ( 3 \pi - 4 ) } { 7 }$
C) $\frac { 16 ( 3 \pi - 4 ) } { 5 }$
D) $\frac { 16 ( 3 \pi - 4 ) } { 3 }$
E) $\frac { 16 ( 3 \pi - 4 ) } { 11 }$

Quizplus · Accepted Answer

The volume can be found by integrating in cylindrical coordinates. The given equations describe a sphere of radius 2 and a cylinder shifted along the x-axis. The intersection of these two shapes in the first octant (due to $x \geq 0$) can be integrated using cylindrical coordinates. The correct calculation leads to the volume $\frac { 16 ( 3 \pi - 4 ) } { 9 }$.

Using Cylindrical Coordinates, the Volume of the Solid Enclosed By x