area between the curves y=4-x^2 and y=x-8
The curves intersect at (-4,-12) and (3,-5)
So, the area is just the sum of all the tiny rectangles with width dx, and height the distance between the curves:
A = ∫[-4,3] (4-x^2)-(x-8) dx
= ∫[-4,3] -x^2-x+12 dx
= -1/3 x^3 - 1/2 x^2 + 12x [-4,3]
= 343/6