Find the volume of the solid obtained by rotating the region bounded by $y=25x-5x^2$ and $y=0$ about the line $x=10.$

Solution

The following figures show the region and a cylindrical shell formed by rotation about the line $x=10.$

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It has radius $10-x,$ circumference $2(10-x),$ and height $-$ The volume of the given solid is

$V={\displaystyle \underset{0}{\overset{5}{}}}2(10-x)(25x-5x^2)dx$

$=2{\displaystyle \underset{0}{\overset{5}{}}}(5x^3-75x^2+250x)dx$

$=2[{]}_0^5$

$=$