Question

Find the volume of the solid by using cylindrical shells?

Answer 1

`color(blue)[V=2piint_0^1(y)(sqrty-y^3)*dy=(2*pi)/5 (unite)^3]`

Explanation:

the volume by cylindrical shell method when the curve rotating about the `"x-axis"` is given by:

`color(red)[V=2piint_a^b(y)(x_2-x_1)*dy]`

`x_2=+-sqrty`
but the area enclosed by the curve lies in the first quadrant.

so `x_2=sqrty`

`x_1=y^3`

lets find the interval of integral.

`sqrty=y^3` square both sides

`y^6=y rArr y^6-y=0`

`y*(y^5-1)=0`

`y=0 and y=1`

the interval of the integral `y in [0,1]`

now let set up the integral:

`V=2piint_0^1(y)(sqrty-y^3)*dy`

`V=2piint_0^1(y^(3/2)-y^4)*dy=[-(2*pi*(y^5-2*y^(5/2)))/5]_0^1`

`=(2*pi)/5 (unite)^3`

show below the region revolving (shaded region) :

show the link below that will help you to understand how to find the volume by cylindrical shell method:
cylindrical shell method