Finding the volume bound by the curves $y = 0 $ and $y = \sqrt x$ and $\space x = 4$ about $x = 0$

Question

this is the picture I came up with. So I am revolving around the y-axis. Therefore, I should be using the function $x = y^2$ since that is the cross-section. Moreover, I will be integrating from $0 to 2$. So, I have the integration $\int_{0}^{2}\pi y^4 = {\pi \over 5}32$. However, when I tried this online, the website got $128\pi\over 5$. What am I doing wrong?

Answer 1

Note that

$$\pi r^2 h - \frac{32\pi}{5}=\pi 4^2 \cdot (2) - \frac{32\pi}{5}=\frac{128\pi}{5}$$

Use the volume of the cylinder subtract the region that you found.