Question

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

Answer 1

The answer is `V=4`

`y<=2 <=> 2>=sqrt(2x) <=> 0<=x<=2`

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

`V(f,alpha)=alphaint_Df(x)^2dx`, where `D` is the domain of `f`

So here we have

`V=piint_0^2 2xdx=pix^2|_0^2 =4pi`