Find the area of the region.

Question

y = x² − 2x,  y = 3x + 6

Answer 1

The curves intersect when x² - 2x = 3x + 6.

x² - 5x - 6 = 0

(x - 6)(x + 1) = 0 So, x = -1 or 6.

Draw a diagram. You should see that the region is bounded above by the line with equation y = 3x + 6 and below by the parabola with equation y = x² - 2x.

Area = ∫_{(-1 to 6)} [3x + 6 - (x² - 2x)]dx = ∫_{(-1 to 6)} [-x² + 5x + 6]dx =

[(-1/3)x³ + (5/2)x² + 6x]_{(-1 to 6)} = (-72 + 90 + 36) - (1/3 + 5/2 - 6) = 60 - 1/3 - 5/2 = 57.1666