Notes: `y =x , y = 0 , y=4, x = 5` Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the...

Monday, 29 February 2016

$\displaystyle{y}={x},{y}={0},{y}={4},{x}={5}$ Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the...

For the region bounded by $\displaystyle{y}={x}$ , $\displaystyle{y}={0}$ , $\displaystyle{y}={4}$ and $\displaystyle{x}={5}$ and revolved about the line $\displaystyle{x}={5}$ , we may also apply the Shell method. we are to use two sets of vertical rectangular strips parallel to the line x=5 (axis of revolution). In this case, we need two sets of rectangular strip since the upper bound of the rectangular strip before and after x=4 differs.

We follow the formula: $\displaystyle{V}={\int_{{a}}^{{b}}}{2}\pi$ * radius*height*thickness

where:

radius (r)= distance of the rectangular strip to the axis of revolution

height (h) = length of the rectangular strip

thickness = width of the rectangular strip as $\displaystyle{\left.{d}{x}\right.}$ or $\displaystyle{\left.{d}{y}\right.}$ .

As shown on the attached file, both rectangular strip has:

$\displaystyle{r}={5}-{x}$

$\displaystyle{h}={y}_{{{a}{b}{o}{v}{e}}}-{y}_{{{b}{e}{l}{o}{w}}}$

thickness $\displaystyle={\left.{d}{x}\right.}$

For the rectangular strip representing the bounded region from x=0 to x=4, we may let:

$\displaystyle{h}={x}-{0}={x}$

For the rectangular strip representing the bounded region from $\displaystyle{x}={4}$ to $\displaystyle{x}={5}$ , we may let:

$\displaystyle{h}={4}-{0}={4}$

Plug-in the values correspondingly, we get:

$\displaystyle{V}={\int_{{0}}^{{4}}}{2}\pi\cdot{\left({5}-{x}\right)}{\left({x}\right)}{\left.{d}{x}\right.}+{2}\pi{\int_{{4}}^{{5}}}{\left({5}-{x}\right)}{\left({4}\right)}{\left.{d}{x}\right.}$

$\displaystyle{V}={2}\pi{\int_{{0}}^{{4}}}{\left({5}-{x}\right)}{\left({x}\right)}{\left.{d}{x}\right.}+{2}\pi{\int_{{4}}^{{5}}}{\left({5}-{x}\right)}{\left({4}\right)}{\left.{d}{x}\right.}$

For the first integral, we solve it as:

$\displaystyle{2}\pi{\int_{{0}}^{{4}}}{2}\pi\cdot{\left({5}{x}-{{x}}^{{2}}\right)}{\left.{d}{x}\right.}$

$\displaystyle={2}\pi\cdot{\left[{5}\frac{{{x}}^{{2}}}{{2}}-\frac{{{x}}^{{3}}}{{3}}\right]}{{\mid}_{{0}}^{{4}}}$

$\displaystyle={2}\pi\cdot{\left[{\left({5}\frac{{{\left({4}\right)}}^{{2}}}{{2}}-\frac{{{\left({4}\right)}}^{{3}}}{{3}}\right)}-{\left({5}\frac{{{\left({0}\right)}}^{{2}}}{{2}}-\frac{{{\left({0}\right)}}^{{3}}}{{3}}\right)}\right]}$

$\displaystyle={2}\pi\cdot{\left[{\left({40}-\frac{{64}}{{3}}\right)}-{\left({0}-{0}\right)}\right]}$

$\displaystyle={2}\pi\cdot{\left[\frac{{56}}{{3}}\right]}$

$\displaystyle=\frac{{{112}\pi}}{{3}}$

For the second integral, we solve it as:

$\displaystyle{2}\pi{\int_{{4}}^{{5}}}{2}\pi\cdot{\left({20}-{4}{x}\right)}{\left.{d}{x}\right.}$

$\displaystyle={2}\pi\cdot{\left[{20}{x}-{4}\frac{{{x}}^{{2}}}{{2}}\right]}{{\mid}_{{4}}^{{5}}}$

$\displaystyle={2}\pi\cdot{\left[{20}{x}-{2}{{x}}^{{2}}\right]}{{\mid}_{{4}}^{{5}}}$

$\displaystyle={2}\pi\cdot{\left[{\left({20}{\left({5}\right)}-{2}{{\left({5}\right)}}^{{2}}\right)}-{\left({20}{\left({4}\right)}-{2}{{\left({4}\right)}}^{{2}}\right)}\right]}$

$\displaystyle={2}\pi\cdot{\left[{\left({100}-{50}\right)}-{\left({80}-{32}\right)}\right]}$

$\displaystyle={2}\pi\cdot{\left[{50}-{48}\right]}$

$\displaystyle={2}\pi\cdot{\left[{2}\right]}$

$\displaystyle={4}\pi$

Combing the two results, we get:

$\displaystyle{V}=\frac{{{112}\pi}}{{3}}+{4}\pi$

$\displaystyle{V}=\frac{{{124}\pi}}{{3}}$ or $\displaystyle{129.85}$ ( approximated value).

We will get the same result whether we use Disk Method or Shell Method for the given bounded region on this problem.

Notes

Monday, 29 February 2016

$\displaystyle{y}={x},{y}={0},{y}={4},{x}={5}$ Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the...

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Monday, 29 February 2016

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the...

No comments:

Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

$\displaystyle{y}={x},{y}={0},{y}={4},{x}={5}$ Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the...

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?