Answer
$A=\frac{262}{3}$
Work Step by Step
$f(x)=-x^2+x+20$ $and$ $g(x)=x^2-5x$
$When$ $f(x)=g(x),$ $x=-2,5$
$A=\int_{4}^5(f(x)-g(x))dx+\int_5^8(g(x)-f(x))dx$
$A=\int_{4}^5(-2x^2+6x+20)dx+\int_5^8(2x^2-6x+20)dx$
$A=[-\frac{2x^3}{3}+3x^2+20x]_{4}^5+[\frac{2x^3}{3}-3x^2-20x]_5^8$
$A=[-\frac{250}{3}+75+100]-[-\frac{128}{3}+48+80]+[\frac{1024}{3}-192-160]-[\frac{250}{3}-75-100]$
$A=[\frac{275}{3}]-[\frac{256}{3}]+[-\frac{32}{3}]-[-\frac{275}{3}]$
$A=\frac{262}{3}$
