Notes: `y = -x + 1` Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

Saturday, 9 September 2017

$\displaystyle{y}=-{x}+{1}$ Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

For the region bounded by $\displaystyle{y}=-{x}+{1}$ revolve about the x-axis, we can also apply the Shell Method using a horizontal rectangular strip parallel to the axis of revolution (x-axis).We may follow the formula for Shell Method as:

$\displaystyle{V}={\int_{{a}}^{{b}}}{2}\pi$ * 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 $\displaystyle{\left.{d}{y}\right.}$ ...

$\displaystyle{V}={\int_{{a}}^{{b}}}{2}\pi$ * 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, the rectangular strip has:

$\displaystyle{r}={y}$

$\displaystyle{h}={f{{\left({y}\right)}}}$ or $\displaystyle{h}={x}_{{2}}-{x}_{{1}}$

$\displaystyle{h}={\left({1}-{y}\right)}-{0}={1}-{y}$

Note: $\displaystyle{y}=-{x}+{1}$ can be rearrange into $\displaystyle{x}={1}-{y}$ .

Thickness $\displaystyle={\left.{d}{y}\right.}$

Boundary values of y: $\displaystyle{a}={0}$ to $\displaystyle{b}={1}$ .

Plug-in the values on to the formula $\displaystyle{V}={\int_{{a}}^{{b}}}{2}\pi$ * radius*height*thickness, we get:

$\displaystyle{V}={\int_{{0}}^{{1}}}{2}\pi\cdot{y}\cdot{\left({1}-{y}\right)}\cdot{\left.{d}{y}\right.}$

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

$\displaystyle{V}={2}\pi{\int_{{0}}^{{1}}}{y}\cdot{\left({1}-{y}\right)}\cdot{\left.{d}{y}\right.}$

Simplify: $\displaystyle{V}={2}\pi{\int_{{0}}^{{1}}}{\left({y}-{{y}}^{{2}}\right)}{\left.{d}{y}\right.}$

Apply basic integration property: $\displaystyle\int{\left({u}-{v}\right)}{\left.{d}{y}\right.}=\int{\left({u}\right)}{\left.{d}{y}\right.}-\int{\left({v}\right)}{\left.{d}{y}\right.}$ to be able to integrate them separately using Power rule for integration: $\displaystyle\int{{y}}^{{n}}{\left.{d}{y}\right.}=\frac{{{y}}^{{{n}+{1}}}}{{{n}+{1}}}$ .

$\displaystyle{V}={2}\pi\cdot{\left[{\int_{{0}}^{{1}}}{\left({y}\right)}{\left.{d}{y}\right.}-{\int_{{0}}^{{1}}}{\left({{y}}^{{2}}\right)}{\left.{d}{y}\right.}\right]}$

$\displaystyle{V}={2}\pi\cdot{\left[\frac{{{y}}^{{2}}}{{2}}-\frac{{{y}}^{{3}}}{{3}}\right]}{{\mid}_{{0}}^{{1}}}$

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

$\displaystyle{V}={2}\pi\cdot{\left[\frac{{{\left({1}\right)}}^{{2}}}{{2}}-\frac{{{\left({1}\right)}}^{{3}}}{{3}}\right]}-{2}\pi\cdot{\left[\frac{{{\left({0}\right)}}^{{2}}}{{2}}-\frac{{{\left({0}\right)}}^{{3}}}{{3}}\right]}$

$\displaystyle{V}={2}\pi\cdot{\left[\frac{{1}}{{2}}-\frac{{1}}{{3}}\right]}-{2}\pi\cdot{\left[{0}-{0}\right]}$

$\displaystyle{V}={2}\pi\cdot{\left[\frac{{1}}{{6}}\right]}-{2}\pi\cdot{\left[{0}\right]}$

$\displaystyle{V}=\frac{{{2}\pi}}{{6}}-{0}$

$\displaystyle{V}=\frac{{{2}\pi}}{{6}}$ or $\displaystyle\frac{\pi}{{3}}$

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

Notes

Saturday, 9 September 2017

$\displaystyle{y}=-{x}+{1}$ Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

No comments:

Post a Comment

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

Saturday, 9 September 2017

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

No comments:

Post a Comment

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

$\displaystyle{y}=-{x}+{1}$ Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

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