# The Fundamental Theorem of Calculus Questions?

Background Information: One of the most important ideas that Green discussed in his Essay is the connection between what happens within a body and the properties of that body’s surface. He realized that, because the boundary of an object is one dimension lower than the interior, the connection can be represented by transforming k−dimensional integrals to simpler integrals involving a (k−1)−dimensional object. In fact, the origin of Green’s Theorem goes to the very heart of Calculus: the Fundamental Theorem.

The Question: Write out the Fundamental Theorem of Calculus. How might we see this as reducing a k−dimensional integral to a (k − 1) dimensional integral?

The work so Far: The Fundamental Theorem of Calculus states that when taking the anti derivative of an integrand you can evaluate the integral by taking the endpoints of the integral and subtracting using the anti derivative. The formula for the Fundamental Theorem of Calculus is as follows: $$\int^{x=b}_{x=a} \, f(x) \, dx = F(b) - F(a).$$

Well, we know that the boundary of $$[a,b]$$ is $$\{a,b\}$$. Then, in some sense, $$\int_a^b f = \int_{\partial[a,b]} F = \int_{\{a,b\}} F.$$ Put another way, $$\int_{[a,b]} \partial f = \int_{\partial[a,b]} f.$$ In particular, this is true if we define $$\int_{\{a,b\}} f := f(b) - f(a)$$. Such a definition is reasonable if you think about what an integral represents: the area (2d concept) under a curve (1d) vs the distance (like length, the 1d analogue of area) between points (0d). In this sense, the FTOC is just a specific instance of Stokes' theorem: $$\int_{\partial\Omega} \omega = \int_\Omega d\omega.$$