Question

How do you find the volume of a rotated region bounded by `y=sqrt(x)`, `y=3`, the y-axis about the y-axis?

Answer 1

Circular cross sections of the bounded region have an area
`pi x^2`
or, since `x = y^2`
`A(y) = pi y^4`

For a thin enough slice, `Delta y`, the volume of the slice approaches
`S(y) = Delta y * A(y)`

and the volume of the bounded region would be
`V(y) = int_0^3 pi y^4 dy`

`= pi int_0^3 y^4 dy`

`= pi (y^5)/5 |_0^3`

`= 48.6 pi`

Answer 2

Try this: