1. Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.

Question

y = 2x³, y = 2x, for x ≥ 0

A) 4/15 π

B) 2/15 π

C) 8/15 π

D) 4/5 π

Answer 1

The region is in the first quadrant, bounded above by y = 2x and bounded below by y = 2x³. By the shell method, the volume of the solid of revolution is 2π ∫ (height)(radius)(thickness). Here, radius = x, height = 2x - 2x³, and thickness = dx. The x bounds are 0 and 1 since the two curves intersect at x = 0 and x = 1 (for x ≥ 0). Thus

Volume = 2π ∫₀¹ (2x - 2x³)x dx = 2π • 2 ∫₀¹ (x² - x⁴) dx = 4π (x³/3 - x⁵/5)|₀¹ = 4π (1/3 - 1/5) = 8π/15

The answer is therefore C).