Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 25, 0 ≤ z ≤ 1, and a hemispherical cap defined by x^2 + y^2 + (z − 1)^2 = 25, z ≥ 1. For the vector field F = (zx + z^2y + 3y, z^3yx + 8x, z^4x^2), compute ∬_M(∇ × F) · dS in any way you like.
∬_M(∇ × F) · dS =