Blog: `y = 1 - x^2, y = 0` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch...

Friday, March 22, 2013

$\displaystyle{y}={1}-{{x}}^{{2}},{y}={0}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch...

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{y}={1}-{{x}}^{{2}},{y}={0}$ , 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 endpoints, hence you need to solve the following equation:

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

You may evaluate the volume

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

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

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

$\displaystyle{V}=\pi\cdot{\left({x}-{2}\frac{{{x}}^{{3}}}{{3}}+\frac{{{x}}^{{5}}}{{5}}\right)}{{\mid}_{{-{1}}}^{{1}}}$

$\displaystyle{V}=\pi\cdot{\left({1}-{2}\cdot\frac{{{1}}^{{3}}}{{3}}+\frac{{{1}}^{{5}}}{{5}}-{\left(-{1}\right)}+{2}\cdot\frac{{{{\left(-{1}\right)}}^{{3}}}}{{3}}-\frac{{{{\left(-{1}\right)}}^{{5}}}}{{5}}\right)}$

$\displaystyle{V}=\pi\cdot{\left({1}-\frac{{2}}{{3}}+\frac{{1}}{{5}}+{1}-\frac{{2}}{{3}}+\frac{{1}}{{5}}\right)}$

$\displaystyle{V}=\pi\cdot{\left({2}-\frac{{4}}{{3}}+\frac{{2}}{{5}}\right)}$

$\displaystyle{V}=\pi\cdot\frac{{{30}-{20}+{6}}}{{15}}$

$\displaystyle{V}=\frac{{{16}\pi}}{{15}}$

Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{y}={1}-{{x}}^{{2}},{y}={0}$ , about x axis, yields $\displaystyle{V}=\frac{{{16}\pi}}{{15}}.$

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Friday, March 22, 2013

$\displaystyle{y}={1}-{{x}}^{{2}},{y}={0}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch...

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