Chapter 13 — Key Takeaways
The Definite Integral in One Sentence
The definite integral $\int_a^b f(x)\,dx$ is the limit of Riemann sums — slice the interval, sum the rectangle areas, take the limit — and it equals the signed area between $f$ and the $x$-axis. It is a single number, the total accumulation of $f$ over $[a,b]$, and it exists for every continuous function (§13.4, §13.12).
1. The Area Problem (§13.1–13.2)
- Geometry gives areas of rectangles, triangles, and circles, but not the area under a curved top like $y = x^2$.
- The fix: approximate with rectangles. Cut $[a,b]$ into $n$ strips of width $\Delta x$, top each strip with a rectangle, and add.
- For a monotonic function the left and right sums bracket the truth: on an increasing $f$, left sums underestimate and right sums overestimate (and vice versa). Their gap is $R_n - L_n = (f(b) - f(a))\,\Delta x \to 0$ (§13.2).
2. Riemann Sums and Sigma Notation (§13.3)
- A regular partition sets $\Delta x = \dfrac{b-a}{n}$ and $x_i = a + i\,\Delta x$.
- A Riemann sum is $\displaystyle S_n = \sum_{i=1}^n f(x_i^*)\,\Delta x$, with sample point $x_i^*$ in the $i$-th strip.
- Standard sample points: left $x_i^* = a + (i-1)\Delta x$; right $x_i^* = a + i\,\Delta x$; midpoint $x_i^* = a + (i - \tfrac12)\Delta x$.
- Watch the index. Right-endpoint sums run $i = 1$ to $n$; left-endpoint sums run $i = 0$ to $n-1$. An off-by-one corrupts every digit (§13.3).
3. The Integral as a Limit (§13.4–13.5)
$$\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x.$$
- For continuous $f$ the limit exists and is independent of the sample-point choice.
- The symbol's anatomy: $\int$ is an elongated "S" for summa; $f(x)$ is the integrand; $dx$ is the limit of $\Delta x$; $a, b$ are the limits of integration; the result is a number, and $x$ is a dummy variable.
- Worked from scratch: $\displaystyle\int_0^1 x^2\,dx = \lim_{n\to\infty}\frac{1}{n^3}\sum_{i=1}^n i^2 = \lim_{n\to\infty}\left(\frac13 + \frac{1}{2n} + \frac{1}{6n^2}\right) = \frac13$, using $\sum i^2 = \frac{n(n+1)(2n+1)}{6}$ (§13.5).
4. Signed Area (§13.6)
- The integral counts area above the axis as positive and below as negative: $$\int_a^b f\,dx = (\text{area above}) - (\text{area below}).$$
- This "net" reading is a feature: it lets the integral report net change (e.g., a particle returning to start has $\int v\,dt = 0$).
- For total (unsigned) area, integrate $|f|$.
- Shortcut: the integral of an odd function over a symmetric interval $[-c, c]$ is $0$.
5. Evaluating by Geometry (§13.7)
When the region is a known shape, read the integral off the picture — no FTC needed:
| Integrand | Region | Value |
|---|---|---|
| $\int_a^b c\,dx$ | rectangle | $c(b-a)$ |
| $\int_0^2 (3x+1)\,dx$ | trapezoid | $\frac{1+7}{2}\cdot 2 = 8$ |
| $\int_{-1}^1 |x|\,dx$ | two triangles | $1$ |
| $\int_0^1 \sqrt{1-x^2}\,dx$ | quarter disk | $\frac{\pi}{4}$ |
Caution: geometry gives signed area. For $\int_0^{2\pi}\sin x\,dx$ the two humps cancel to $0$, not twice one hump (§13.7).
6. Properties of the Definite Integral (§13.8)
| Property | Statement |
|---|---|
| Linearity | $\int_a^b(\alpha f + \beta g) = \alpha\int_a^b f + \beta\int_a^b g$ |
| Additivity | $\int_a^c f + \int_c^b f = \int_a^b f$ |
| Reversal | $\int_a^b f = -\int_b^a f$ |
| Zero width | $\int_a^a f = 0$ |
| Comparison | $f \le g \implies \int_a^b f \le \int_a^b g$ |
| Bounding box | $m \le f \le M \implies m(b-a) \le \int_a^b f \le M(b-a)$ |
Each is a statement about Riemann sums in the limit, and each has a one-line picture.
7. Average Value and the MVT for Integrals (§13.11)
- Average value: $\displaystyle \overline{f} = \frac{1}{b-a}\int_a^b f(x)\,dx$ — the height of the rectangle on $[a,b]$ with the same area as the region under $f$.
- MVT for Integrals: if $f$ is continuous on $[a,b]$, there is a $c \in [a,b]$ with $f(c) = \overline{f}$, i.e. $\int_a^b f = f(c)(b-a)$.
- Why: the bounding box puts $\overline{f}$ between $\min f$ and $\max f$; the Intermediate Value Theorem (Chapter 4) then says $f$ attains it. Continuity does the real work — and this is exactly the tool Chapter 14 uses to prove the FTC.
8. Existence: When Is $f$ Integrable? (§13.12)
- Continuous functions on a closed bounded interval are integrable: uniform continuity squeezes upper and lower sums together.
- Piecewise-continuous bounded functions (finitely many jumps) are integrable too.
- The Dirichlet function (1 on rationals, 0 on irrationals) is not integrable: its upper sum is always 1, lower sum always 0. The repair, Lebesgue integration, belongs to a later course.
9. Numerical Integration (§13.13)
- Midpoint rule: $M_n = \Delta x\sum f(\overline{x}_i)$ — samples strip centers; errors self-cancel.
- Trapezoidal rule: $T_n = \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)] = \frac12(L_n + R_n)$.
- Error scaling: rectangles $O(h)$; midpoint and trapezoid $O(h^2)$. Doubling $n$ roughly quarters the midpoint/trapezoid error.
- Accuracy assumes a smooth integrand; near a kink or singularity the rates degrade (Chapter 17 handles the singular case).
10. The Anchor — Area Under the Normal Curve (§13.14)
- The standard normal density $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ has no elementary antiderivative (Liouville).
- Probabilities are areas: $P(a \le X \le b) = \int_a^b \phi(x)\,dx$, computed by numerical Riemann sums — the source of every $z$-table value.
- $P(-1 \le X \le 1) \approx 0.6827$ — the "68%" of the empirical rule — from rectangles alone.
- This is the book's recurring anchor: revisited in Chapter 14 (why $\Phi' = \phi$) and Chapter 23 (Taylor-series evaluation).
Looking Ahead: The Fundamental Theorem (Ch. 14)
Everything here was computed the hard way — sums, geometry, limits — on purpose. Chapter 14 proves the Fundamental Theorem of Calculus: $$\int_a^b f(x)\,dx = F(b) - F(a), \qquad F' = f,$$ collapsing the sum-of-squares limit of §13.5 into the one-liner $\int_0^1 x^2\,dx = \frac{1^3}{3} - 0 = \frac13$. The integral you built in this chapter is the object that theorem is about (§13.15).
Common Errors to Avoid
- Off-by-one in sums: right endpoints run $i=1$ to $n$; left endpoints run $i=0$ to $n-1$ (§13.3).
- Confusing distance with displacement: $\int v\,dt$ is signed displacement; total distance is $\int |v|\,dt$, split at every sign change (§13.10).
- Reading geometry as unsigned area: below-axis regions subtract; check the sign of the integrand first (§13.6–13.7).
- Reaching for an antiderivative: the FTC shortcut does not exist yet — this chapter evaluates by sums, geometry, and limits only.
- Trusting numerical rules near singularities: $O(h^2)$ accuracy assumes smoothness; a midpoint sum across an asymptote can be confidently wrong (§13.13).
Connections
- Back to Chapter 3 (limits) and Chapter 4 (continuity, IVT, EVT, uniform continuity): these underwrite the existence of the integral and the MVT for Integrals.
- Back to Chapter 12 (antiderivatives): the antiderivatives you found there become the shortcut in Chapter 14.
- Forward to Chapter 14 (FTC): turns Riemann sums into subtraction.
- Forward to Chapters 15–18: integration techniques and applications (areas, volumes, work) all rerun the slice–sum–limit pattern.
- Forward to Chapter 23: Taylor series evaluate the bell-curve area to any precision.
- Far forward to Chapters 32, 35–37: double, triple, line, and surface integrals are higher-dimensional echoes of this same definition.