Question

Does anyone understand why we are using double integrals for Gauss's Law? Why aren't we just using single integrals?

see page 3 of:
http://ocw.mit.edu/courses/physics/8-02sc-physics-ii-electricity-and-magnetism-fall-2010/electric-fields/gauss-law/MIT8_02SC_notes5.pdf

If you don't want to click the link, it is the pdf from the Gauss's Law section of Electric Fields.

Answer 1

We use double integrals because it's a surface. When you define the surface, dA will fall into two variables (like dxdy or drdΘ) which requires 2 integrals.

Answer 2

mathematically we compute area of a surface using double integration i.e if the surface is defined in the x-y plane the area is integration over dx and dy .