I've attached a diagram to show what I've calculated.
The area which is painted green in my diagram is a circle with radius R minus a smaller circle with radius r:
From y = 9 - x² you get: x = √(9 - y)
R = 3, (constant)
r = 3 - (√(9-y))
The outer circle has the area A_(oc) = π * R² that means A_(oc) = 9π
The inner circle has the area A_(ic) = π * (3 - (√(9-y))² that means A_(ic) = π*(y - 6(√(9-y) +18)
The green area is the difference between the 2 circles:
A = 9π - π*(y - 6(√(9-y) +18) = π*(y + 6(√(9-y) -9)
The volume you are looking for is
V = ∫(π*(y + 6(√(9-y) -9))dy from(0) to(9). Use substitution method. You'llget:
V = (-4π(9-y)^(3/2)+π/2 y² - 9πy) from(0) to(9).
V = 135/2 π
V ≈ 212.06
Your result is 13.5 π. So I guess you made a typo somewhere in your calculations.