`y = e^(-x^2), y = 0, x = 0, x = 1` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the...

Wednesday, January 20, 2016

$\displaystyle{y}={{e}}^{{-{{x}}^{{2}}}},{y}={0},{x}={0},{x}={1}$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the...

The shell has the radius x, the cricumference is $\displaystyle{2}\pi\cdot{x}$ and the height is $\displaystyle{{e}}^{{-{{x}}^{{2}}}}$ , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

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

You need to use substitution method to solve the integral, such that:

$\displaystyle-{{x}}^{{2}}={u}\Rightarrow-{2}{x}{\left.{d}{x}\right.}={d}{u}\Rightarrow{x}{\left.{d}{x}\right.}=-\frac{{{d}{u}}}{{2}}$

$\displaystyle{V}={2}\pi\cdot{\int_{{{u}_{{1}}}}^{{{{u}}^{{2}}}}}{{e}}^{{u}}\cdot\frac{{-{d}{u}}}{{2}}$

$\displaystyle{V}=-\pi\cdot{{e}}^{{u}}{{\mid}_{{{u}_{{1}}}}^{{{{u}}^{{2}}}}}$

$\displaystyle{V}=-\pi\cdot{{e}}^{{-{{x}}^{{2}}}}{{\mid}_{{0}}^{{1}}}$

$\displaystyle{V}=-\pi\cdot{\left({{e}}^{{-{{1}}^{{2}}}}-{{e}}^{{-{{0}}^{{2}}}}\right)}$

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

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

Hence, evaluating the volume, using the method of cylindrical shells, yields $\displaystyle{V}=\frac{{{\left({{e}}^{{2}}-{1}\right)}\cdot\pi}}{{e}}.$

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Wednesday, January 20, 2016

$\displaystyle{y}={{e}}^{{-{{x}}^{{2}}}},{y}={0},{x}={0},{x}={1}$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the...

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