Question

Find the area bounded by the curve x²=y+2 and the line x+y=0.

Answer 1

Let us find the x points of intersection of two graphs $y_1(x) = x^2 - 2$ and $y_2(x) = - x$ by solving corresponding quadratic equation $x^2 - 2 = -x$. It has roots $x = -2 , x = 1$.

Hence, the area between two curves is:

$S = \int_{-2}^1 (-x - (x^2 - 2 )) d x = (-\frac{x^2}{2} - \frac{x^3}{3} + 2 x)|_{-2}^1 = \frac{9}{2}$.