Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = 5x − x^2, y = 4; about x = 1

Answer 1

Volume, V = 2π∫_a^br(x)h(x)dx

r(x) = x-1 and h(x) =5x-x²-4

To get a and b, 5x-x² = 4 --> x = 1,4

V = 2π∫₁⁴(x-1)(5x-x²-4)dx = 2π∫₁⁴ -x³+6x²+9x+4dx

V = 2π[-x⁴/4+2x³-9x²/2+4x]₁⁴ = 27π/2=13.5π