# How do you find the volume of the solid obtained by rotating the region bounded by the curves x=y-y^2 and the y axis rotated around the y-axis?

Jul 29, 2015

$\frac{\pi}{30}$

#### Explanation:

The curve represents a horizontal parabola as seen in the picture

The volume of the solid so generated would be(consider an elementary strip of length and thickness $\delta$y. If it is rotated about x axis its volume would be $\pi {x}^{2} \mathrm{dy}$. The volume of the solid generated by rotating the whole shaded region would be

${\int}_{0}^{1} \pi {x}^{2} \mathrm{dy}$

=${\int}_{0}^{1} \pi {\left(y - {y}^{2}\right)}^{2} \mathrm{dy}$

=${\int}_{0}^{1} \pi \left({y}^{2} - 2 {y}^{3} + {y}^{4}\right) \mathrm{dy}$

=$\pi {\left({y}^{3} / 3 - 2 {y}^{4} / 4 + {y}^{5} / 5\right)}_{0}^{1}$

=$\frac{\pi}{30}$