Find the rate of heat flow inward across the cylindrical surface

The temperature at the point (x, y, z) in a substance with conductivity
K = 8.5 is u(x, y, z) = 3y^2 + 3z^2.
Find the rate of heat flow inward across the cylindrical surface
y^2 + z^2 = 5, 0 ≤ x ≤ 3.

Updated post:
The heat flow is given by F(x,y,z) = -K∇u = -8.5 <0, 6y, 6z> = -51 <0, y, z>.

Using cylindrical coordinates with x playing the usual role of z, we have
R(u, v) = with u in [0, 3] and v in [0, 2π].

Since R_u x R_v = <0, -√5 cos v, -√5 sin v>, the rate of heat flow equals
∫∫s F · dS
= ∫∫s -51 <0, √5 cos v, √5 sin v · <0, -√5 cos v, -√5 sin v> dA
= ∫(u = 0 to 3) ∫(v = 0 to 2π) 205 dv du
= 205 * 3 * 2π
= 1230π.

I hope this helps!