how to find the area of the part paraboloid x=y^2+z^2 that lies inside the cylinder y^2+z^2 = 9?

Using Cartesian coordinates (with x playing the usual role of z):
A = ∫∫ √(1 + (x_y)^2 + (x_z)^2) dA
....= ∫∫ √(1 + (2y)^2 + (2z)^2) dA
....= ∫∫ √(1 + 4(y^2 + z^2)) dA

Since the region of integration is inside y^2 + z^2 = 9, convert to polar coordinates:
∫(r = 0 to 3) ∫(θ = 0 to 2π) √(1 + 4r^2) * (r dθ dr)
= ∫(r = 0 to 3) 2πr (1 + 4r^2)^(1/2) dr
= 2π * (1/8)(2/3)(1 + 4r^2)^(3/2) {for r = 0 to 3}
= (π/6) (37^(3/2) - 1).

I hope this helps!

you mean find the volume of a part of paraboloid which is easy to do

∫_(-3)^3 ∫_[-sqrt(9 - (y^2))]^sqrt(9 - (y^2)) ∫_0^[(y^2) + (z^2)] dx dz dy = 81π/2

∫_(0)^(2π) ∫_0^3 ∫_0^(r^2) r dx dr dθ = 81π/2