Question

Find the area of the region enclosed by the parabola $x^2=y$ and the line $y=x+2$.

Answer 1

The region enclosed by the parabola $x^2=y$, the line $y=x+2$, and $x$-axis is represented by the shaded region $OABCO$ as
The point of intersection of the parabola $x^2=y$ and the line $y=x+2$ is $A(-1,1)$.
$\therefore AreaOABCO=Area\left(BCA\right)+AreaCOAC$
$={\int }_{-2}^{-1}(x+2)dx+{\int }_{-1}^0{x}^{2}dx$
$={[\frac{x^2}{2}+2x]}_{-2}^{-1}+{\left[\frac{x^3}{3}\right]}_{-1}^0$
$=[\frac{(-1{)}^2}{2}+2(-1)-\frac{(-2{)}^2}{2}-2(-2\left)\right]+[0-\frac{(-1{)}^3}{3}]$
$=[\frac{1}{2}-2-2+4+\frac{1}{3}]$
$=\frac{5}{6}$ sq. units