Use spherical coordinates. Evaluate ,where B is the ball with center the origin and radius 5.
E) None of these

Question

Use spherical coordinates. Evaluate  ,where B is the ball with center the origin and radius 5.
E) None of these

Quizplus · Accepted Answer

In spherical coordinates, the volume of a ball centered at the origin with radius $r$ is $\frac{4}{3}\pi r^3$. Using this formula with $r=11eaae29_f43e_57d3_87fe_a5c34c6c1bee_TB2067_11$, we have that the volume of the ball is $\frac{4}{3}\pi (11eaae29_f43e_57d3_87fe_a5c34c6c1bee_TB2067_11)^3$. To find the value of the expression given, we need to find the volume of the portion of the ball inside the sphere with radius $11eaae29_f43e_30c1_87fe_9b1480406d8f_TB2067_11$. Since the point $(0,0,11eaae29_f43e_30c1_87fe_9b1480406d8f_TB2067_11)$ is on the surface of the inner sphere, we can use symmetry to say that the portion of the ball inside the inner sphere can be obtained by taking the portion of a sphere with radius $11eaae29_f43e_30c1_87fe_9b1480406d8f_TB2067_11$ that lies in the first octant, and multiplying by $8$. The volume of this portion of the sphere is equal to $\frac{1}{8}\cdot \frac{4}{3}\pi (11eaae29_f43e_30c1_87fe_9b1480406d8f_TB2067_11)^3$. Dividing this expression by the volume of the full ball yields $\frac{1}{8}$, which when multiplied by the volume of the full ball gives the volume of the portion of the ball inside the inner sphere, which is $\frac{1}{8}\cdot \frac{4}{3}\pi (11eaae29_f43e_57d3_87fe_a5c34c6c1bee_TB2067_11)^3$. Thus, the value of the expression given is $\frac{1}{8}\cdot \frac{4}{3}\pi (11eaae29_f43e_57d3_87fe_a5c34c6c1bee_TB2067_11)^3=\boxed{	extbf{(C)}\ 11eaae29_f43e_7ee6_87fe_7d22185e915d_TB2067_11}$.

Use Spherical Coordinates