Use the shell method to find the volume of the solid below the surface formed by revolving the curve $z = 8 \sin y ( 0 \leq y \leq \pi ) \text { about the } z \text {-axis and above the } x y \text {-plane. }$
A) $V = 8 \pi ^ { 3 }$
B) $V = 16 \pi$
C) $V = 16 \pi ^ { 2 }$
D) $V = 8 \pi$
E) $V = 16 \pi ^ { 3 }$

Question

Use the shell method to find the volume of the solid below the surface formed by revolving the curve $z = 8 \sin y ( 0 \leq y \leq \pi ) \text { about the } z \text {-axis and above the } x y \text {-plane. }$ 
A) $V = 8 \pi ^ { 3 }$
B) $V = 16 \pi$
C) $V = 16 \pi ^ { 2 }$
D) $V = 8 \pi$
E) $V = 16 \pi ^ { 3 }$

Quizplus · Accepted Answer

The volume of the solid formed by revolving the curve $z = 8 \sin y$ about the z-axis can be found using the shell method. The formula for the volume using the shell method is $V = 2\pi \int_a^b yf(y) dy$, where $f(y) = 8 \sin y$, $a = 0$, and $b = \pi$. Substituting these values into the formula gives $V = 2\pi \int_0^\pi y(8 \sin y) dy = 16\pi \int_0^\pi y \sin y dy$. Integrating by parts or looking up the integral, we find that the integral equals $\pi^2$, thus $V = 16\pi^2$.

Use the Shell Method to Find the Volume of the Solid