# 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

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

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### 1 Answer

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` .**