Notes: `pi(int_0^1(y^4 - y^8)dy)` Each integral represents the volume of a solid. Describe the solid.

Friday, 31 March 2017

$\displaystyle\pi{\left({\int_{{0}}^{{1}}}{\left({{y}}^{{4}}-{{y}}^{{8}}\right)}{\left.{d}{y}\right.}\right)}$ Each integral represents the volume of a solid. Describe the solid.

The formula provided represents the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{x}={{y}}^{{2}},{x}={{y}}^{{4}}$ , the lines x = 0, x = 1, about y axis.

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{1}}}{\left({{\left({{y}}^{{2}}\right)}}^{{2}}-{{\left({{y}}^{{4}}\right)}}^{{2}}\right)}{\left.{d}{y}\right.}$

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{1}}}{{y}}^{{4}}{\left.{d}{y}\right.}-\pi\cdot{\int_{{0}}^{{1}}}{{y}}^{{8}}{\left.{d}{y}\right.}$

$\displaystyle{V}={\left(\pi\cdot\frac{{{y}}^{{5}}}{{5}}-\pi\cdot\frac{{{y}}^{{9}}}{{9}}\right)}{{\mid}_{{0}}^{{1}}}$

$\displaystyle{V}=\pi\cdot{\left(\frac{{{1}}^{{5}}}{{5}}-\frac{{{1}}^{{9}}}{{9}}-\frac{{{0}}^{{5}}}{{5}}+\frac{{{0}}^{{9}}}{{9}}\right)}$

$\displaystyle{V}=\pi\cdot{\left(\frac{{1}}{{5}}-\frac{{1}}{{9}}\right)}$

$\displaystyle{V}=\ldots$

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{1}}}{\left({{\left({{y}}^{{2}}\right)}}^{{2}}-{{\left({{y}}^{{4}}\right)}}^{{2}}\right)}{\left.{d}{y}\right.}$

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{1}}}{{y}}^{{4}}{\left.{d}{y}\right.}-\pi\cdot{\int_{{0}}^{{1}}}{{y}}^{{8}}{\left.{d}{y}\right.}$

$\displaystyle{V}={\left(\pi\cdot\frac{{{y}}^{{5}}}{{5}}-\pi\cdot\frac{{{y}}^{{9}}}{{9}}\right)}{{\mid}_{{0}}^{{1}}}$

$\displaystyle{V}=\pi\cdot{\left(\frac{{{1}}^{{5}}}{{5}}-\frac{{{1}}^{{9}}}{{9}}-\frac{{{0}}^{{5}}}{{5}}+\frac{{{0}}^{{9}}}{{9}}\right)}$

$\displaystyle{V}=\pi\cdot{\left(\frac{{1}}{{5}}-\frac{{1}}{{9}}\right)}$

$\displaystyle{V}=\pi\cdot\frac{{{9}-{5}}}{{45}}$

$\displaystyle{V}=\frac{{{4}\pi}}{{45}}$

Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{x}={{y}}^{{2}},{x}={{y}}^{{4}}$ , the lines $\displaystyle{x}={0},$ $\displaystyle{x}={1}$ , about y axis, yields $\displaystyle{V}=\frac{{{4}\pi}}{{45}}$ .