Area of integrals between curves

Question

Find the area of the region enclosed by the curves  𝑥=𝑦^3−𝑦^2 and x=2y (see diagram, desmos).  Show and explain your work.

Answer 1

y³ - y² = 2y; y = 0, y = -1, y = 2

1. Area A = ∫_-1⁰(y³ - y² - 2y)dy + ∫₀²(2y + y² - y³)dy = (y⁴/4 - y³/3 - y²)_-1⁰ + (y² + y³/3 - y⁴/4)₀² = - 1/4 - 1/3 + 1 + 4 + 8/3 - 4 = 37/12