Notes: `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...

Tuesday, 8 November 2016

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

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

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

...

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

Notes