Notes: `y = xsqrt(4-x^2) , y=0` Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.

Sunday, 17 July 2016

$\displaystyle{y}={x}\sqrt{{{4}-{{x}}^{{2}}}},{y}={0}$ Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.

The region bounded by $\displaystyle{y}={x}\sqrt{{{4}-{{x}}^{{2}}}}$ and $\displaystyle{y}={0}$ revolved about the x-axis is shown on the attached image.We may apply the Disk method using a rectangular strip perpendicular to the axis of revolution. As shown on the attached image, the thickness of the rectangular strip is "dx" with a vertical orientation perpendicular to the x-axis (axis of revolution).

We follow the formula for the Disk method: $\displaystyle{V}={\int_{{a}}^{{b}}}{A}{\left({x}\right)}{\left.{d}{x}\right.}$ where disk's base area is $\displaystyle{A}=\pi{{r}}^{{2}}$ with $\displaystyle{r}={y}={f{{\left({x}\right)}}}$ .

Note: r = length of the rectangular strip. We may apply $\displaystyle{r}={y}_{{{a}{b}{o}{v}{e}}}-{y}_{{{b}{e}{l}{o}{w}}}$ .

Then $\displaystyle{r}={\left({x}\sqrt{{{4}-{{x}}^{{2}}}}\right)}-{0}={x}\sqrt{{{4}-{{x}}^{{2}}}}$ .

The boundary values of x are $\displaystyle{a}=-{2}$ to $\displaystyle{b}={2}$ .

Plug-in the $\displaystyle{f{{\left({x}\right)}}}$ and the boundary values to integral formula, we get:

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

Simplify:

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

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

Apply basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$

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

Apply basic integration property: $\displaystyle\int{\left({u}-{v}\right)}{\left.{d}{x}\right.}=\int{\left({u}\right)}{\left.{d}{x}\right.}-\int{\left({v}\right)}{\left.{d}{x}\right.}$ .

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

Apply Power rule for integration: $\displaystyle\int{{x}}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}}^{{{n}+{1}}}}{{{n}+{1}}}$ .

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

Apply definite integration formula: $\displaystyle{\int_{{a}}^{{b}}}{f{{\left({y}\right)}}}{\left.{d}{y}\right.}={F}{\left({b}\right)}-{F}{\left({a}\right)}$ .

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

$\displaystyle{V}=\pi\cdot{\left[\frac{{32}}{{3}}-\frac{{32}}{{5}}\right]}-\pi\cdot{\left[\frac{{-{32}}}{{3}}-\frac{{-{32}}}{{5}}\right]}$

$\displaystyle{V}=\pi\cdot{\left[\frac{{160}}{{15}}-\frac{{96}}{{15}}\right]}-\pi\cdot{\left[\frac{{-{160}}}{{15}}-\frac{{-{96}}}{{15}}\right]}$

$\displaystyle{V}=\pi\cdot{\left[\frac{{64}}{{15}}\right]}-\pi\cdot{\left[\frac{{-{64}}}{{15}}\right]}$

$\displaystyle{V}=\frac{{{64}\pi}}{{15}}-\frac{{-{64}\pi}}{{15}}$

$\displaystyle{V}=\frac{{{64}\pi}}{{15}}+{64}\frac{\pi}{{15}}$

$\displaystyle{V}=\frac{{{128}\pi}}{{15}}$ or $\displaystyle{26.81}$ (approximated value)

Notes

Sunday, 17 July 2016

$\displaystyle{y}={x}\sqrt{{{4}-{{x}}^{{2}}}},{y}={0}$ Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.

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

Sunday, 17 July 2016

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

No comments:

Post a Comment

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

$\displaystyle{y}={x}\sqrt{{{4}-{{x}}^{{2}}}},{y}={0}$ Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.

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