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Mathematics

OpenStudy (anonymous):

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

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OpenStudy (dumbcow):

|dw:1314347044031:dw| 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. \[V = \pi \int\limits_{0}^{1}(y-y^{2})^{2} dy\] \[ = \pi \int\limits\limits_{0}^{1}(y^{2}-2y^{3}+y^{4})dy\] \[=\pi [\frac{1}{3}y^{3}-\frac{1}{2}y^{4}+\frac{1}{5}y^{5}] \left(\begin{matrix}1 \\ 0\end{matrix}\right)\] \[=\frac{\pi}{30}\]

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