# How do you find the volume of a rotated region bounded by y=sqrt(x), y=3, the y-axis about the y-axis?

Mar 13, 2015

Circular cross sections of the bounded region have an area
$\pi {x}^{2}$
or, since $x = {y}^{2}$
$A \left(y\right) = \pi {y}^{4}$

For a thin enough slice, $\Delta y$, the volume of the slice approaches
$S \left(y\right) = \Delta y \cdot A \left(y\right)$

and the volume of the bounded region would be
$V \left(y\right) = {\int}_{0}^{3} \pi {y}^{4} \mathrm{dy}$

$= \pi {\int}_{0}^{3} {y}^{4} \mathrm{dy}$

$= \pi \frac{{y}^{5}}{5} {|}_{0}^{3}$

$= 48.6 \pi$

Mar 13, 2015

Try this: