Integrate the given function over the surface. G(x,y,z)=z on S;{(x,y,z)│y^2+z^2=4,1<x<4}?

Trasfer it into standard form: G(x,y,z) = x

S: x^2 + y^2 = 4

z = 1~4


Set z = h(x,y)

=> G(x,y,z) = h(x,y) - z

=> ∂G/∂x = ∂h/∂x - ∂z/∂x

=> ∂G/∂x = ∂h/∂x - 0 = ∂h/∂x

=> ∂G/∂y = ∂h/∂y



dS = √[1+(∂h/∂x)^2+(∂h/∂y)^2]*dxdy

= √[1+(∂G/∂x)^2+(∂G/∂y)^2]*dxdy

= √(1+1+0)*dxdy

= √2*dxdy


∫∫G(x,y,z)dS

= ∫∫√2*x*dy*dx

= ∫√2xydx

= ∫√2x√(4-x^2)dx

= -∫√2/2 * √(4-x^2) * d(4-x^2)

= -√2/2 * 2/3 * (4-x^2)^1.5 ;;; x = 0~2

= - √2/3 * [(4-4)^1.5 - (4-0)^1.5]

= √2/3 * 2^3

= 8√2/3

= Answer