Question

A region r is bounded by y=x and y = x^2. set up the integral to find the volume v of the solid formed by rotating r around the x-axis and then find the volume.

Answer 1

the area between 2 curves, f(x) and g(x) when f(x) is above g(x) and they intersect at a and b and around x axis is

`A=\pi \int\limits^a_b {f(x)^2-g(x)^2} \, dx`

alrighty, find where they intersect
x=x^2 at x=0 and x=1

and x^2 is above so


`A=\pi \int\limits^1_0 {(x^2)^2-(x)^2} \, dx`

`A=\pi \int\limits^1_0 {x^4-x^2} \, dx`

`A=\pi[(x^5)/(5)-(x^3)/(3)]\limits^1_0`

`A=\pi(((1^5)/(5)-(1^3)/(3))-((0^5)/(5)-(0^3)/(3)))`

`A=\pi(((1)/(5)-(1)/(3))-0)`

`A=\pi((3)/(15)-(5)/(15))`

`A=\pi((-2)/(15))`

`A=(-2\pi)/(15)`

the area is
`(-2\pi)/(15)`