Lecture · 18 min · intermediate

The integral

Area, accumulation, and the fundamental theorem

Courant · the definite integral and the FTC

If the derivative answers “how fast?”, the integral answers “how much, in total?” Area under a curve is the prototype, but the same construction accumulates any density: distance from speed, mass from density, probability from a density function. The fundamental theorem says these two questions are inverses.

On the board today

—See the definite integral as a limit of Riemann sums.
—Use antiderivatives to evaluate definite integrals.
—State both halves of the fundamental theorem.

Fundamental theorem of calculus

\frac{d}{d x} \int_{a}^{x} f (t) d t = f (x), \int_{a}^{b} f = F (b) - F (a)

If $F^{'} = f$ . Accumulation, then evaluation.

A Riemann sum chops $[a, b]$ into $n$ pieces, picks a height $f (x_{i}^{*})$ on each, and adds areas $f (x_{i}^{*}) Δ x$ . The integral is the limit as the mesh goes to $0$ . Signed area: below the axis counts negative. The FTC is the reason we bother with antiderivatives at all — it turns an infinite sum into two evaluations of $F$ .

Worked example · Area under a parabola

Compute $\int_{0}^{2} x^{2} d x$ .

1.An antiderivative of $x^{2}$ is $\frac{1}{3} x^{3}$ .
2.Evaluate: $\frac{1}{3} (8) - \frac{1}{3} (0) = \frac{8}{3}$ .

The area is $\frac{8}{3}$ .

Pause the lecture

If $F (x) = \int_{0}^{x} sin (t^{2}) d t$ , what is $F^{'} (x)$ ? Do not try to “compute the integral.”

Common mistake. Forgetting the constant when writing the general antiderivative, or dropping the evaluation $F (b) - F (a)$ and reporting $F (x)$ as the answer to a definite integral.

Practice this lecture

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