`y = x^3, y = 8, x = 0` Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the...

Wednesday, June 2, 2010

The volume of the solid obtained by rotating about x-axis by using cylindrical shell method is

$\displaystyle{V}={\int_{{a}}^{{b}}}{2}\pi{y}{f{{\left({y}\right)}}}{\left.{d}{y}\right.}$

The given information is

The curves

$\displaystyle{y}={{x}}^{{3}}=\gt{x}={{y}}^{{\frac{{1}}{{3}}}}$

$\displaystyle{y}={8}$ , $\displaystyle{x}={y}={0}$ and rotation is about y-axis

$\displaystyle\therefore{V}={\int_{{0}}^{{8}}}{2}\pi{y}{\left[{{y}}^{{\frac{{1}}{{3}}}}-{0}\right]}{\left.{d}{y}\right.}$

= $\displaystyle{2}\pi{\int_{{0}}^{{8}}}{{y}}^{{\frac{{4}}{{3}}}}{\left.{d}{y}\right.}$

= $\displaystyle{2}\pi\frac{{3}}{{7}}{{y}}^{{\frac{{7}}{{3}}}}{{\mid}_{{0}}^{{8}}}$

= $\displaystyle\frac{{{6}\pi}}{{7}}\cdot{{8}}^{{\frac{{7}}{{3}}}}$

= $\displaystyle\frac{{{6}\pi}}{{7}}\cdot{{2}}^{{7}}$

= $\displaystyle\frac{{{768}\pi}}{{7}}$

$\displaystyle\therefore$ The required volume is $\displaystyle\frac{{{768}\pi}}{{7}}$

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