the region bounded by y=0 and y=kx-x^2 is revolved around the y-axis to form a solid whose volume is 10.

Question

what are the exact possible values of k? Hint: use cylindrical shells

Answer 1

The curves intersect when kx - x² = 0

 

                                     x(k-x) = 0   So, x = 0 or k

 

Volume of a typical shell = 2π(radius)(height)dx = 2πx(kx-x²)dx

 

Volume of solid = 2π∫(kx² - x³)dx from x = 0 to x = k

 

                      = 2π[kx³/3 - (1/4)x⁴] from x = 0 to x = k

 

                      = (π/6)k⁴

 

So, if volume = 10, then (π/6)k⁴ = 10

 

   k⁴ = 60/π

 

   k = ±(60/π)^¼