Find the volume of the solid formed by rotating the region enclosed by x=0, x=1, y=0, y= 3+x^4 about the y-axis. Use integrals to find the volume. Do it the way that you would find the volume in...

Find the volume of the solid formed by rotating the region enclosed by

x=0, x=1, y=0, y= 3+x^4 about the y-axis.

Use integrals to find the volume. Do it the way that you would find the volume in calculus 2

Asked on by chocobob

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lfryerda | High School Teacher | (Level 2) Educator

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There are several ways to find the volume of revolution.  The simplest is to divide the volume into two regions.  The first region is the cylinder from y=0 to y=3, with radius 1.  This cylinder has volume of

`V=pi1^2 (3)`

`=3pi`

The second region is from y=3 to x=1, which has y=4.  The volume in this region is found using hollow discs, with outside radius 1, inside radius x and height dy.  This means the volume of each slice is

`dV=pi(1-x^2)dy`    solve for `x^2` in `y=3+x^4` which gives `x^2=sqrt{y-3}` so the volume slice is

`=pi(1-sqrt{y-3})dy`

which may be integrated from 3 to 4.

`V=pi int_3^4(1-sqrt{y-3})dy`   split into two integrals.  The first is:

`pi int_3^4 dy`

`=pi y |_3^4`

`=pi`

The second integral is 

`-pi int_3^4 sqrt{y-3}dy`   let `u=y-3` then `du=dy` and the limits are 0 and 1

`=-pi int_0^1u^{1/2}du`

`=-pi 2/3 u^{3/2}|_0^1`

`=-{2pi}/3`

This means the total volume is 

`V=3pi+pi-2/3pi`

`={10pi}/3`

The volume of the solid is `{10pi}/3` .

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