Question

Find the area of the region bounded by the line y=3x and y= x^3 + 2x^2?

and

find the area of the region bounded by the curve y=e^2x -3e^x + 2 and the x-axis?

Answer 1

Where do they intersect?
3x = x^3 + 2 x^2

x^3 + 2 x^2 - 3 x = 0
x (x^2 + 2x -3) = 0
x (x+3)(x-1) = 0
intersect at x = -3, 0, 1
from x = -3 to x = 0 the cubic is above the line so in that domain the area is integral [ x^3 + 2 x^2 - 3 x] dx
from x = 0 to x = 1 the line is above the cubic so in that domain the area is
integral [ 3 x - x^3 - 2 x^2] dx