Question

Find the area of the region enclosed by the curves x= y^2-2 and x=y.

Answer 1

1) Limits of integration:

$$y^2-2=y$$

$$D=(-1)^2-4(-2)=9=3^2$$

$$y_1=\frac{1-3}{2}=-1$$

$$y_2=\frac{1+3}{2}=2$$

2)

$$A=\int_{-1}^2 (y-(y^2-2))dy$$

$$A=\left(-\frac{y^3}{3}+\frac{y^2}{2}+2y\right)_{-1}^2$$

$$A=\left(-\frac{2^3}{3}+\frac{2^2}{2}+2(2)\right)-\left(-\frac{(-1)^3}{3}+\frac{(-1)^2}{2}+2(-1)\right)$$

$$A=\frac{9}{2}=4.5$$