Thursday, February 9, 2012

The Fundamental Theorem of Calculus

This theorem is so important that it's called the Fundamental Theorem of Calculus.  Why is it so important?  First, it provides a link between differentiation and integration.  Second, it gives a way to easily evaluate definite integrals, bypassing the use of Riemann sums.

The theorem comes in two parts.  Let's call them FTC1 and FTC2.


Let f(x) be a continuous function on [a,b], and let \( a \leq x \leq b \).  Then
\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)

Let f(x) be a continuous function on [a,b], and let F(x) be an antiderivative of f(x).  Then
\int_{a}^{b} f(x) dx = F(b) - F(a)
Since \( F'(x) = f(x) \), we can rewrite this as
\int_{a}^{b} \frac{d}{dx}F(x) dx = F(b) - F(a)

If you look at the last equation and the equation from FTC1, you can see how differentiation and integration are inverse operations in some sense.

FTC2 itself shows that you can easily evaluate a definite integral if you know an antiderivative of the integrand.  The alternative is to go through laborious calculations with Riemann sums.  Using FTC2, you trade off tedious calculations for memorizing antiderivatives or looking them up--basically a time-memory trade-off.  This is what makes calculus so powerful as a tool.

Geometric Interpretation



There are higher-dimensional versions of this theorem.  There are 2-dimensional versions called Green's theorem and Stoke's theorem.  The 3-dimensional version is called the Divergence Theorem or Gauss' theorem.  There is also the general n-dimensional Stoke's theorem.