Answer
$A = \frac{9}{2}$
Work Step by Step
Find points of intersection by setting the functions equal to each other.
$f(y) = y^{2}$ and $g(y) = y + 2$ therefore
$y^{2}$ = y + 2
$y^{2}$ - y - 2 = 0
(y - 2)(y + 1) = 0
y = -1, 2
The functions intersect at y = -1, 2. Therefore, they are the endpoints of the integral.
Integrate right minus left to find the area. Remember to integrate in terms of dy.
$A =\int_{-1}^2[g(y) -f(y)]dy$
$=\int_{-1}^2[(y + 2) -y^{2}]dy$
$= [\frac{y^{2}}{2} + 2y - \frac{y^{3}}{3}]_{-1}^2$
$= [\frac{4}{2} + 4 - \frac{8}{3}] - [\frac{1}{2} - 2 + \frac{1}{3}]$
$= \frac{3}{2} + 6 -3$
$A = \frac{9}{2}$
