Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-$$y = 4 \sqrt { x } , y = 0 , x = 1$$
A) $\frac { 8 } { 5 } \pi$
B) $8 \pi$
C) $\frac { 16 } { 5 } \pi$
D) $20 \pi$

Question

Quizplus · Accepted Answer

The volume of the solid generated by revolving the region about the y-axis can be found using the shell method. The formula for the shell method is $V = 2\pi \int_{a}^{b} x f(x) dx$, where $f(x)$ is the function being revolved, and $a$ and $b$ are the limits of integration. Here, $f(x) = 4\sqrt{x}$, and the limits of integration are from $x = 0$ to $x = 1$. Thus, the volume is $V = 2\pi \int_{0}^{1} x(4\sqrt{x}) dx = 2\pi \int_{0}^{1} 4x^{3/2} dx$. Integrating, we get $V = 2\pi [\frac{8}{5}x^{5/2}]_{0}^{1} = \frac{16}{5}\pi$.

Use the Shell Method to Find the Volume of the Solid