Applications of Integration

Question

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by y=x⁶, y=1, and the y-axis about the line y=−3.

Volume =

Answer 1

Limits of integration: x⁶ = 1; x = 1 because x ≥ 0.

Using disk-washer method we have: volume V = π∫₀¹[(1 - (-3)²) - (x⁶ - (-3))²]dx = π∫₀¹(7 - x¹² - 6x⁶)dx =

π(7x - x¹³/13 - 6x⁷/7)₀¹ = π(7 - 1/13 - 6/7) = 552π/91= 19.0567