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, $y$.
A) $V ( y ) = \frac { \pi } { 2 \sqrt { 2 } } y ^ { 1 / 2 }$
B) $V ( y ) = \frac { 3 \pi } { 2 \sqrt { 2 } } y ^ { 3 / 2 }$
C) $V ( y ) = \frac { \pi } { 9 } y ^ { 9 }$
D) $V ( y ) = \frac { 2 \pi } { 3 \sqrt { 2 } } y ^ { 3 / 2 }$

Question

Quizplus · Accepted Answer

To find the volume of water in the tank as a function of the water depth, we need to integrate the cross-sectional area of the tank with respect to the depth.

First, let's find the equation of the curve when rotated about the $y$-axis. Using the formula for the volume of revolution, we have

$$V = \int_{0}^{y}\pi x^{2}dy$$

where $x$ is a function in terms of $y$. Rearranging the given equation, we get $x = (y/2)^{1/4}$. Substituting, we have

$$V = \int_{0}^{y} \pi \left(\frac{y}{2}\right)^{1/2} dy$$

$$V = \frac{2\pi}{3}\left(\frac{y}{2}\right)^{3/2}$$

$$V = \frac{2\pi}{3\sqrt{2}} y^{3/2}$$

Therefore, the best choice is D.

A Water Tank Is Formed by Revolving the Curve y=2x4y = 2 x ^ { 4 }y=2x4

A Water Tank Is Formed by Revolving the Curve $y = 2 x ^ { 4 }$