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

$A=\pi \int\limits^1_0 {(x^2)^2-(x)^2} \, dx$
$A=\pi \int\limits^1_0 {x^4-x^2} \, dx$
$A=\pi[\frac{x^5}{5}-\frac{x^3}{3}]\limits^1_0$
$A=\pi((\frac{1^5}{5}-\frac{1^3}{3})-(\frac{0^5}{5}-\frac{0^3}{3}))$
$A=\pi((\frac{1}{5}-\frac{1}{3})-0)$
$A=\pi(\frac{3}{15}-\frac{5}{15})$
$A=\pi(\frac{-2}{15})$
$A=\frac{-2\pi}{15}$

the area is $\frac{-2\pi}{15}$