Solve the problem.
-A water tank is formed by revolving the curve $y = 2 x ^ { 4 }$ about the $y$-axis. Find the volume of water in the tank as a function of the water depth, $\mathrm { y }$.
A) $V ( y ) = \frac { 2 \pi } { 3 \sqrt { 2 } } y ^ { 3 / 2 }$
B) $V ( y ) = \frac { \pi } { 2 \sqrt { 2 } } y ^ { 1 / 2 }$
C) $V ( y ) = \frac { 3 \pi } { 2 \sqrt { 2 } } y ^ { 3 / 2 }$
D) $V ( y ) = \frac { \pi } { 9 } y ^ { 9 }$

Question

Quizplus · Accepted Answer

The volume of the water tank can be found using the method of disks or washers, integrating with respect to $y$. The equation $y = 2x^4$ is solved for $x$ to find the radius of a disk at a height $y$, giving $x = (y/2)^{1/4}$. The area of a disk is $\pi r^2 = \pi (y/2)^{1/2}$, and integrating this with respect to $y$ from 0 to $y$ gives the volume $V(y) = \int_0^y \pi (y/2)^{1/2} dy$, which simplifies to $V(y) = \frac{2\pi}{3\sqrt{2}}y^{3/2}$.

Solve the Problem y=2x4y = 2 x ^ { 4 }y=2x4

Solve the Problem $y = 2 x ^ { 4 }$