Fundamental Theorem of Integral Calculus

Building a Riemann-type sum and taking a limit every time an integral needs to be evaluated (Section 4.1) is reliable but slow. The Fundamental Theorem of Integral Calculus removes that need entirely.

Statement and proof. Let $f$ be continuous on $[a,b]$ and suppose $\int f(x)\,dx=g(x)+c$ (i.e. $g$ is any one primitive of $f$). Then

The arbitrary constant $c$ always cancels, so

— evaluate any one primitive at the upper limit, subtract its value at the lower limit. In $\int_a^b f(x)\,dx$, $a$ is called the lower limit and $b$ the upper limit of integration.

This single formula is why definite integration is normally done by finding an indefinite integral (a primitive) first and then substituting the two limits, rather than by the limit-of-a-sum construction — the limit-of-sum method of Section 4.1 is the theoretical justification, and the Fundamental Theorem is the practical shortcut built on top of it. All of the properties in the next sub-section, and essentially every solved example and exercise in this chapter after this point, use this two-step evaluate-and-subtract idea. …