How do you find the volume of the region left of y = sqrt(2x) and below y = 2 rotated about the y-axis?

May 24, 2015

The answer is $V = 4$

$y \le 2 \iff 2 \ge \sqrt{2 x} \iff 0 \le x \le 2$

So, you can use Guldino's formula for the volume of a solid of revolution:

$V \left(f , \alpha\right) = \alpha {\int}_{D} f {\left(x\right)}^{2} \mathrm{dx}$, where $D$ is the domain of $f$

So here we have

$V = \pi {\int}_{0}^{2} 2 x \mathrm{dx} = \pi {x}^{2} {|}_{0}^{2} = 4 \pi$