Question

the region bounded by the graph f(x)=x(2-x) and the x axis is revolved about the y axis. Find the volume of the solid. I did the integral using the shell method, but the answer wasn't correct.

Answer 1

Why don't you show your work?

Using shells, try

v = ∫[0,2] 2πrh dx
where r = x and h=y=2x-x^2
v = ∫[0,2] 2πx(2x-x^2) dx
= 8π/3

Using washers, we have

v = ∫[0,1] π(R^2-r^2) dy
where r = 1-√(1-y^2) and R = 1+√(1-y^2)
v = ∫[0,1] π((1+√(1-y^2))^2-(1-√(1-y^2))^2) dy
which is a little more complicated.

Answer 2

My bad. Using discs, the integral is

v = ∫[0,1] π(R^2-r^2) dy
where r = 1-√(1-y) and R = 1+√(1-y)
v = ∫[0,1] π((1+√(1-y))^2-(1-√(1-y))^2) dy
= ∫[0,1] 4π√(1-y) dy
= 8π/3

Answer 3

i did get 8pi/3 but it wasn't one of the choices. Maybe there is a mistake in the question