Notes: `y = 4x - x^2, y = x` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

Friday, 18 March 2016

$\displaystyle{y}={4}{x}-{{x}}^{{2}},{y}={x}$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

The shell has the radius $\displaystyle{x}$ , the cricumference is $\displaystyle{2}\pi\cdot{x}$ and the height is $\displaystyle{4}{x}-{{x}}^{{2}}-{x}$ , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

$\displaystyle{V}={2}\pi\cdot{\int_{{{x}_{{1}}}}^{{{x}_{{2}}}}}{x}\cdot{\left({3}{x}-{{x}}^{{2}}\right)}{\left.{d}{y}\right.}$

You need to evaluate the endpoints $\displaystyle{x}_{{1}}$ and $\displaystyle{x}_{{2}}$ , such that:

$\displaystyle{4}{x}-{{x}}^{{2}}={x}\Rightarrow{3}{x}-{{x}}^{{2}}={0}\Rightarrow{x}{\left({3}-{x}\right)}={0}\Rightarrow{x}={0}{\quad\text{and}\quad}{3}-{x}\ldots$

$\displaystyle{V}={2}\pi\cdot{\int_{{{x}_{{1}}}}^{{{x}_{{2}}}}}{x}\cdot{\left({3}{x}-{{x}}^{{2}}\right)}{\left.{d}{y}\right.}$

You need to evaluate the endpoints $\displaystyle{x}_{{1}}$ and $\displaystyle{x}_{{2}}$ , such that:

$\displaystyle{4}{x}-{{x}}^{{2}}={x}\Rightarrow{3}{x}-{{x}}^{{2}}={0}\Rightarrow{x}{\left({3}-{x}\right)}={0}\Rightarrow{x}={0}{\quad\text{and}\quad}{3}-{x}={0}\Rightarrow{x}={3}$

$\displaystyle{V}={2}\pi\cdot{\int_{{0}}^{{3}}}{x}\cdot{\left({3}{x}-{{x}}^{{2}}\right)}{\left.{d}{y}\right.}$

$\displaystyle{V}={2}\pi\cdot{\left({\int_{{0}}^{{3}}}{3}{{x}}^{{2}}{\left.{d}{x}\right.}-{\int_{{0}}^{{3}}}{{x}}^{{3}}{\left.{d}{x}\right.}\right)}$

Using the formula $\displaystyle\int{{x}}^{{n}}{\left.{d}{x}\right.}=\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}$ yields:

$\displaystyle{V}={2}\pi\cdot{\left({3}\frac{{{x}}^{{3}}}{{3}}-\frac{{{x}}^{{4}}}{{4}}\right)}{{\mid}_{{0}}^{{3}}}$

$\displaystyle{V}={2}\pi\cdot{\left({{x}}^{{3}}-\frac{{{x}}^{{4}}}{{4}}\right)}{{\mid}_{{0}}^{{3}}}$

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

$\displaystyle{V}={2}\pi\cdot\frac{{{{3}}^{{3}}}}{{4}}$

$\displaystyle{V}=\frac{{{27}\pi}}{{2}}$

Hence, evaluating the volume, using the method of cylindrical shells, yields $\displaystyle{V}=\frac{{{27}\pi}}{{2}}.$

Notes

Friday, 18 March 2016

$\displaystyle{y}={4}{x}-{{x}}^{{2}},{y}={x}$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

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

Friday, 18 March 2016

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

No comments:

Post a Comment

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

$\displaystyle{y}={4}{x}-{{x}}^{{2}},{y}={x}$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

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