How do you find the volume of the solid obtained by rotating the region bounded by the curves y^2=4x, x=0 and y=4 about the y axis?

Feb 16, 2016

$32 \pi$

Explanation:

The graph (before rotation) is shown below
graph{sqrt(4x) [-0.5, 4.5, -0.5, 4.5]}

Slice the solid generated in a manner that is normal to the $x$ axis, each slice having a thickness of $\text{d} x$.

Each slice has volume of

$\text{d"V = pi y^2 "d} x$

$= 4 \pi x \text{d} x$

The total volume is given by

${\int}_{0}^{4} 4 \pi x \text{d} x = {\left[2 \pi {x}^{2}\right]}_{0}^{4}$

$= 32 \pi$