Question

find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
x=y-y^2, x=0; about the y-axis

Answer 1

The graph shows the parabola bounded by the y-axis, find volume by summing the cross-section areas. The cross-section is a circle with radius x, shown above. Integrate over y from 0 to 1.
\[V = \pi \int\limits_{0}^{1}(y-y^{2})^{2} dy\]
\[ = \pi \int\limits\limits_{0}^{1}(y^{2}-2y^{3}+y^{4})dy\]
\[=\pi [\frac{1}{3}y^{3}-\frac{1}{2}y^{4}+\frac{1}{5}y^{5}] \left(\begin{matrix}1 \\ 0\end{matrix}\right)\]
\[=\frac{\pi}{30}\]