Consider the region bounded by y=x^2, y=1, and the y-axis, for x≥0. Find the volume of the solid obtained by rotating the region about the line y=−6.

Answer 1

By disk-washer method volume V = π∫₀¹[(1 + 6)² - (x² + 6)²]dx = π∫₀¹(13 - x⁴ - 12x²)dx = π(13x - x⁵/5 - 4x³)₀¹ = π(13 - 1/5 - 4) = 44π/5 = 27.646