Blog: `y = x^3, y = x, x=>0` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line....

Monday, November 28, 2016

$\displaystyle{y}={{x}}^{{3}},{y}={x},{x}\Rightarrow{0}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line....

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{y}={{x}}^{{3}},{y}={x},{x}={0}$ , about x axis, using washer method, such that:

$\displaystyle{V}={\int_{{a}}^{{b}}}{\left({{f}}^{{2}}{\left({x}\right)}-{{g}}^{{2}}{\left({x}\right)}\right)}{\left.{d}{x}\right.},{f{{\left({x}\right)}}}>{g{{\left({x}\right)}}}$

You need to find the next endpoint, since one of them, x = 0 is given. The other endpoint can be evaluated by solving the following equation:

$\displaystyle{{x}}^{{3}}={x}\Rightarrow{{x}}^{{3}}-{x}={0}\Rightarrow{x}{\left({{x}}^{{2}}-{1}\right)}={0}\Rightarrow{x}={0},{x}={1},{x}=-{1}$

You may evaluate the volume

$\displaystyle{V}=\pi\cdot{\int_{{-{1}}}^{{0}}}{\left({{x}}^{{6}}-{{x}}^{{2}}\right)}{\left.{d}{x}\right.}+\pi\cdot{\int_{{0}}^{{1}}}{\left({{x}}^{{2}}-{{x}}^{{6}}\right)}{\left.{d}{x}\right.}$

$\displaystyle{V}=\pi\cdot{\int_{{-{1}}}^{{0}}}{\left({{x}}^{{6}}\right)}{\left.{d}{x}\right.}-\pi\cdot{\int_{{-{1}}}^{{0}}}{{x}}^{{2}}{\left.{d}{x}\right.}+\pi\cdot{\int_{{0}}^{{1}}}{{x}}^{{2}}{\left.{d}{x}\right.}-\pi\cdot{\int_{{0}}^{{1}}}{{x}}^{{6}}{\left.{d}{x}\right.}$

$\displaystyle{V}=\pi\cdot{\left(\frac{{{{x}}^{{7}}}}{{7}}-\frac{{{x}}^{{3}}}{{3}}\right)}{{\left|_{{\left(-{1}\right)}}^{{0}}+\pi\cdot{\left(\frac{{{x}}^{{3}}}{{3}}-\frac{{{x}}^{{7}}}{{7}}\right)}\right|}_{{0}}^{{1}}}$

$\displaystyle{V}=\pi\cdot{\left(\frac{{{{0}}^{{7}}}}{{7}}-\frac{{{0}}^{{3}}}{{3}}-\frac{{1}}{{7}}+\frac{{1}}{{3}}\right)}+\pi\cdot{\left(\frac{{{1}}^{{3}}}{{3}}-\frac{{{1}}^{{7}}}{{7}}-{0}\right)}$

$\displaystyle{V}=\frac{{{4}\pi}}{{21}}+\frac{{{4}\pi}}{{21}}$

$\displaystyle{V}=\frac{{{8}\pi}}{{21}}$

Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{y}={{x}}^{{3}},{y}={x},{x}={0}$ , about x axis, yields $\displaystyle{V}=\frac{{{8}\pi}}{{21}}.$

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Monday, November 28, 2016

$\displaystyle{y}={{x}}^{{3}},{y}={x},{x}\Rightarrow{0}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line....

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