Blog: `pi(int_0^pi(sin(x))dx)` Each integral represents the volume of a solid. Describe the solid.

Tuesday, December 11, 2012

$\displaystyle\pi{\left({\int_{{0}}^{\pi}}{\left({\sin{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}\right)}$ Each integral represents the volume of a solid. Describe the solid.

The formula provided represents the volume of the solid obtained by rotating the region enclosed by the curves $\displaystyle{y}=\sqrt{{{\sin{{x}}}}},{y}={0},$ about y axis, using washer method:

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

You need to find the endpoints by solving the equation:

$\displaystyle\sqrt{{{\sin{{x}}}}}={0}\Rightarrow{\sin{{x}}}={0}\Rightarrow{x}={0},{x}=\pi$

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

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{\pi}}}{\sin{{x}}}{\left.{d}{x}\right.}$

$\displaystyle{V}=\pi\cdot{\left(-{\cos{{x}}}\right)}{{\mid}_{{0}}^{{\pi}}}$

$\displaystyle{V}=\pi\cdot{\left(-{\cos{\pi}}+{\cos{{0}}}\right)}$

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

$\displaystyle{V}={2}\pi$

Hence, evaluating the volume of the solid obtained by rotating the region enclosed by the curves $\displaystyle{y}=\sqrt{{{\sin{{x}}}}},{y}={0}$ , about y axis, using washer method, yields $\displaystyle{V}={2}\pi.$

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Tuesday, December 11, 2012

$\displaystyle\pi{\left({\int_{{0}}^{\pi}}{\left({\sin{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}\right)}$ Each integral represents the volume of a solid. Describe the solid.

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