Answer Happy

The Fundamental Theorem of Calculus always says roughly: Given a region R whose boundary is B, the integral (i.e. a normal integral, line integral, surface integral, multiple integral) of "something" (i.e. a function, 2D vector field, 3D vector field) over B is equal to the integral of "the derivative of that something" (i.e. the regular derivative, the gradient, the curl, or the divergence) over R. The different theorems we saw in chapter 13 are all of this form, its just that the something, the integral, and the derivative all take various different forms. Write out each of the fundamental theorems seen in chapter 13, as well as the standard fundamental theorem from last semester, and in each, say what kind of region R you have, what its boundary B looks like, what types of integrals you're calculating, and what the derivative means.