Find the area of the region

Question

Find the area of the region bounded by the given functions. (fx)=2x, g(x)=4-2x, h(x)=0

and f(x)=x(x^2-3x+3), g(x)=x^2

Answer 1

1. 2x = 4 - 2x; Limits of integration: 4x = 4; x = 1 y = 2; 0 ≤ y ≤ 2; y/2 ≤ x ≤ 2 - y/2

A = ∫₀²(2 - y/2 - y/2dy = (2y - y²/2)₀² = 4 - 2 = 2

2.Intersections: x(x² - 3x + 3) = x²; x = 0, x² - 4x + 3 = 0; x = 1, x = 3

A = ∫₀¹(x³ - 4x² + 3x)dx + ∫₁³(4x² - x³ -3x)dx = (x⁴/4 - 4x³/3 + 3x²/2)₀¹ + (4x³/3 - x⁴/4 - 3x²/2)₁³ =

1/4 - 4/3 + 3/2 + 36 - 81/4 - 27/2 - 4/3 + 1/4 + 3/2 = 5/12 + 8/3 = 37/12