Question

y=x^2-5x,y=0 in y-axis ,,sketch the region R bounded by the graphs of equations and find volume of solid generated if R is revolved about indicated axis

Answer 1

using shells of thickness dx,
v = ∫[0,5] 2πrh dx
where r = x and h=0-y
v = ∫[0,5] 2πx(5x-x^2) dx = 625π/6

using discs of thickness dy,
since y = (x - 5/2)^2 - 25/4,
x = 5/2 ±√(y + 25/4)
v = ∫[-25/4,0] π(R^2-r^2) dy
where R = 5/2 + √(y + 25/4) and r = 5/2 - √(y + 25/4)
v = ∫[-25/4,0] π((5/2 + √(y + 25/4))^2-(5/2 - √(y + 25/4))^2) dy = 625π/6