Area Between Curves Question 2

What is the area of the region between $y=2x^2$ and $y=12-x^2$ ?

$0$

$16$

$32$

$48$

Approach

We need to find the lower and upper limits of integration. Find the intersections of the two curves:

$$ 2x^2=12-x^2 $$ $$ 3x^2 =12 $$ $$ x^2=4 $$ $$ x=2,-2 $$

The area under the curve is:

$$ \int\limits_{-2}^2 (12-x^2)-(2x^2) dx = \int\limits_{-2}^2 12-3x^2 dx $$ $$ = \Big[12x-x^3\Big]_{-2}^2 $$ $$ = (12(2)-(2)^3)-(12(-2)-(-2)^3) $$ $$ (16)-(-16)= \boxed{32} $$