Find the area of the region bounded by the following curves.
$f(x)=x^2+x−27$, $g(x)=−x^2+5x+3$
I know I need to put the equations together and factor them to gets x= -3, 5
$\int^5_{-3}(x^2+x−27)-(x^2+5x+3)$
But after that I am lost, does anyone know what I need to do after to get the answer? It should be round to 3 decimal points.
The region is given below.
You solved correctly the points of intersection. But your formula for the area is not correct. This is how to solved it. $$\begin{align} A&=\int_{-3}^5\big[g(x)-f(x)\big]dx\\\ &=\int_{-3}^5\bigg[(-x^2+5x+3)-(x^2+x-27)\bigg]dx\\ &=\int_{-3}^5(-2x^2+4x+30)dx\\ &=\bigg[-\frac{2x^3}{3}+2x^2+30x\bigg]_{-3}^5\\ &=\bigg(-\frac{250}{3}+50+150\bigg)-\bigg(18+18-90\bigg)\\ &=-\frac{250}{3}+200+54\\ &=\frac{512}{3} \end{align}$$