6.2 Volumes/29

$y^{2}+R=1$

$\int _{0}^{1}\pi (1-y^{2})^{2}-\pi (1-{\sqrt[{31}]{y}})^{2}dy=\pi {\sqrt {\int _{0}^{1}}}1-2y^{2}+y^{4}dy$

$\int _{0}^{1}1-2-{\sqrt[{3}]{y}}+y{\frac {2}{3}}dy=[y-{\frac {2y^{3}}{3}}+{\frac {y^{5}}{5}}]{\bigg |}_{0}^{1}-[y-{\frac {6y}{4}}+{\frac {3y}{5}}]{\bigg |}_{0}^{1}$

$=\pi [(1-{\frac {2}{3}}+{\frac {1}{5}})-(1-{\frac {6}{4}}+{\frac {3}{5}})]=\pi [{\frac {15}{15}}-{\frac {10}{15}}+{\frac {13}{15}})-({\frac {20}{20}}-{\frac {30}{20}}+{\frac {12}{20}})$

$=\pi ({\frac {18}{15}}x{\frac {2}{10}})x3=\pi ({\frac {16}{30}}=\pi ({\frac {13}{30}})=$

$=({\frac {13\pi }{30}})$