Mean Value Theorem in Surface Integrals

[iomtt6076]

I'm reading Div, Grad Curl, and All That, and in coming up with a formula for the divergence, H.M. Schey starts with a small cube centered at (x,y,z), labels the face parallel to the yz-plane as S₁ and calculates

$$\int\int_{S_1}\mathbf{F}\cdot\hat{\mathbf{n}}dS=\int\int_{S_1}F_x(x,y,z)dS$$

He then says that because the cube is small, the integral is equal to F_x evaluated at the center of S₁ times the area of S₁. The justification apparently comes from some mean value theorem that says "the integral of F_x over S₁ is equal to the area of S₁ multiplied by the function evaluated somewhere on S₁."

I tried looking up this theorem to no avail. Could someone point me to a proof or maybe outline why it's true?

Thanks in advance.

 

[Landau]

Take a look at your message, you'll see that the word 'mean value theorem' has become a link. It's essentially the two-dimensional version of
$$\exists\ c\,\in\,(a,b): \int_a^b f(t)\ =\ f(c)\,(b\ -\ a)$$