Solve the problem.
-Evaluate
$$\int _ { - 2 } ^ { 2 } \int _ { - \sqrt { 4 - y ^ { 2 } } } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } - y ^ { 2 } } } d z d x d y$$
by transforming to cylindrical or spherical coordinates.
A) $\frac { 2 \pi } { 3 } \left( 9 - ( 5 ) ^ { 3 / 2 } \right)$
B) $\frac { 2 \pi } { 3 } \left( 27 - ( 5 ) ^ { 3 / 2 } \right)$
C) $\frac { 4 \pi } { 3 } \left( 9 - ( 5 ) ^ { 3 / 2 } \right)$
D) $\frac { 4 \pi } { 3 } \left( 27 - ( 5 ) ^ { 3 / 2 } \right)$

Question

Quizplus · Accepted Answer

The integral represents the volume of the region bounded by a cylinder $x^2 + y^2 = 4$ and a sphere $x^2 + y^2 + z^2 = 9$. Converting to cylindrical coordinates, the limits become $0 \leq r \leq 2$, $0 \leq \theta \leq 2\pi$, and $0 \leq z \leq \sqrt{9 - r^2}$. The volume is $\int_0^{2\pi} \int_0^2 \int_0^{\sqrt{9-r^2}} r \, dz \, dr \, d\theta$, which evaluates to $\frac{2\pi}{3} (27 - 5^{3/2})$.

Solve the Problem By Transforming to Cylindrical or Spherical Coordinates 2π3(9−(5)3/2)\frac { 2 \pi } { 3 } \left( 9 - ( 5 ) ^ { 3 / 2 } \right)32π​(9−(5)3/2)

Solve the Problem By Transforming to Cylindrical or Spherical Coordinates $\frac { 2 \pi } { 3 } \left( 9 - ( 5 ) ^ { 3 / 2 } \right)$