Question

The area bounded by the curve 𝑦 = 3 − 2𝑥 + 𝑥
2 and line 𝑦 = 3 is revolved about the line 𝑦 =3.
Find the volume generated.

Answer 1

the curves intersect at (0,3) and (2,3)
using shells of thickness dx,
v = ∫[0,2] πr^2 dx
where r=3-y
v = ∫[0,2] π(3 - (x^2-2x+3))^2 dx = 16π/15

Using shells of thickness dy, and the symmetry about the line x=1,
v = ∫[2,3] 2πrh dy
where r = 3-y and h = 1-x
v = 2∫[2,3] 2π(3-y)√(y-2) dy = 16π/15