Determine the volume of the solid obtained by rotating the region bounded

Question

 2) Determine the volume of the solid obtained by rotating the region bounded by y=x^2−2x and y=x about the line y=4.

Answer 1

x² - 2x = x; x = 0 or x = 3.

Volume v = π∫₀³[(4- x² + 2x)² - (4 - x)²]dx = π∫₀³(3x - x²)(8 - x² + x)dx = π∫₀³(24x - 4x³ - 5x² + x⁴)dx =

π(12x² - x⁴ - 5x³/3 + x⁵/5)₀³ = π(108 - 81 - 45 + 243/5) = 30.6π