Find the area between the curve y = 8 + 2x - x^2 and the line y = 8 - 2x.

First sketch the curve and the line, noting down where they intersect each axis.area under y = 8 + 2x - x² is given by the integral between 0 and 4 of (8 + 2x - x²) dx.area under line is given by the integral between 0 and 4 of (8-2x) dx. It's easier to do this than using the formula for area of a triangle!!So total area:area = integral between 0 and 4 of (8 + 2x - x²) dx - integral between 0 and 4 of (8-2x) dxarea = integral between 0 and 4 of (8 + 2x - x² - (8-2x))dx Note we can combine the two integrals!!area = integral between 0 and 4 of (4x - x²) dxarea = [2x² - x³/3]⁴₀ = 32/3