Consider the 3-sided region bounded by $x=0$ , $y=4$ and $y=\sqrt{x}$ .What is the exact volume of the solid obtained when this region is rotated around the x-axis?
A) $128 \pi$
B) $8 \pi$
C) $128$
D) $8$

Question

Quizplus · Accepted Answer

The volume of the solid obtained by rotating the given region around the x-axis can be calculated using the disk method. The region is bounded by $x=0$, $y=4$, and $y=\sqrt{x}$. To find the bounds in terms of $x$, we solve $y=\sqrt{x}$ for $x$ when $y=4$, giving $x=16$. Thus, the bounds are $x=0$ to $x=16$. The volume $V$ is given by the integral $\int_{0}^{16} \pi (\sqrt{x})^2 dx = \int_{0}^{16} \pi x dx$, which evaluates to $\frac{1}{2} \pi x^2 |_{0}^{16} = 128\pi$.

Consider the 3-Sided Region Bounded By x=0x=0x=0 y=4y=4y=4 And y=xy=\sqrt{x}y=x​ What Is the Exact Volume of the Solid Obtained When

Consider the 3-Sided Region Bounded By $x=0$ $y=4$ And $y=\sqrt{x}$ What Is the Exact Volume of the Solid Obtained When