Notes: `x = 2sqrt(y), x = 0, y = 9` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified...

Thursday, 17 October 2013

$\displaystyle{x}={2}\sqrt{{{y}}},{x}={0},{y}={9}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified...

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{x}={2}\sqrt{{y}},{x}={0}$ , the line y = 9, about y 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, y = 9 is given. The other endpoint can be evaluated by solving the following equation:

$\displaystyle{2}\sqrt{\ldots}$

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

You need to find the next endpoint, since one of them, y = 9 is given. The other endpoint can be evaluated by solving the following equation:

$\displaystyle{2}\sqrt{{y}}={0}\Rightarrow{4}{y}={0}\Rightarrow{y}={0}$

You may evaluate the volume

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{9}}}{\left({{\left({2}\sqrt{{y}}\right)}}^{{2}}\right)}{\left.{d}{y}\right.}$

$\displaystyle{V}=\pi\cdot{\int_{{0}}^{{9}}}{\left({4}{y}\right)}{\left.{d}{y}\right.}$

$\displaystyle{V}={4}\pi\cdot\frac{{{y}}^{{2}}}{{2}}{{\mid}_{{0}}^{{9}}}$

$\displaystyle{V}={2}\pi\cdot{{y}}^{{2}}{{\mid}_{{0}}^{{9}}}$

$\displaystyle{V}={2}\pi\cdot{\left({{9}}^{{2}}-{{0}}^{{2}}\right)}$

$\displaystyle{V}={2}\pi\cdot{\left({81}\right)}$

$\displaystyle{V}={162}\pi$

Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves $\displaystyle{x}={2}\sqrt{{y}},{x}={0}$ , the line y = 9, about y axis, yields $\displaystyle{V}={162}\pi$ .

Notes

Thursday, 17 October 2013

$\displaystyle{x}={2}\sqrt{{{y}}},{x}={0},{y}={9}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified...

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Is there any personification in "The Tell-Tale Heart"?

Thursday, 17 October 2013

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified...

No comments:

Post a Comment

Is there any personification in &quot;The Tell-Tale Heart&quot;?

$\displaystyle{x}={2}\sqrt{{{y}}},{x}={0},{y}={9}$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified...

Is there any personification in "The Tell-Tale Heart"?