Fundamental Theorem of Calculus

Why a limit of infinitely many sums collapses to one subtraction.

The idea

Theorem (Fundamental Theorem of Calculus).

Let $f$ be continuous on an interval $[a, b]$. Part 1. If $F(x) = \int_a^x f(t) \, dt$ — the amount accumulated from the fixed $a$ out to a varying $x$ — then $F'(x) = f(x)$. Part 2. If $F$ is any function with $F' = f$, then $\int_a^b f(x) \, dx = F(b) - F(a).$

Between them the two parts say that differentiating and integrating undo one another. That is worth a great deal here, because we defined $\int_a^b f$ as a limit of slice sums — an honest definition and a miserable way to compute anything, since you would be adding ever-longer lists forever. Part 2 replaces that entire limit with a single subtraction.

Ways to work on it

• Walkthrough. Each part of the theorem used once.
• Proof. One strip gives Part 1; Part 2 follows from it.
• Practice. Apply Part 2 with the antiderivative supplied.
• Hardest. Part 1, including integrands with no elementary antiderivative.

Not sure where to start? Take the ten-question placement test.