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The Definite Integral

Integration is the second key tool of calculus: it determines the area enclosed between the graph of a function and the $x$-axis. This page follows Walz, Foundations of Mathematics, the first part of Section 6.3: it introduces the definite integral as a limit of rectangular approximations, states when a function is integrable, and collects the basic properties and rules of the definite integral.

Areas Under Simple Graphs

Write $F_a(b)$ for the area that the graph of $f$ encloses with the $x$-axis as $x$ runs over $\lbrack a, b \rbrack$. For the simplest functions this area is elementary geometry:

  • For the constant function $f(x) = 1$, the region is a rectangle: $\int_a^b 1\,dx = F(b) - F(a) = b - a.$
  • For the identity $f(x) = x$, the region is a triangle: $\int_a^b x\,dx = F(b) - F(a) = \frac{b^2}{2} - \frac{a^2}{2}.$

Here $F(b)$ and $F(a)$ are the areas over $\lbrack 0, b \rbrack$ and $\lbrack 0, a \rbrack$, and the enclosed area is their difference — a pattern that will recur throughout.

A Curved Graph: the Parabola

For $f(x) = x^2$ the region is bounded by a curve, so we approximate. Divide $\lbrack 0, b \rbrack$ into $n$ subintervals of width $b/n$ and, on each, replace $f$ by its value at the right endpoint. Summing the resulting rectangles gives a Riemann sum $\mathcal{F}_n = \sum_{j=0}^{n-1} (j+1)^2 \left(\frac{b}{n}\right)^3 = \frac{n(n+1)(2n+1)}{6}\left(\frac{b}{n}\right)^3 = \frac{b^3}{6}\left(1 + \frac{1}{n}\right)\left(2 + \frac{1}{n}\right).$

As the subdivision is refined ($n \to \infty$) the approximation converges: $\lim_{n \to \infty} \mathcal{F}_n = \frac{b^3}{6} \cdot 2 = \frac{b^3}{3}.$

Thus $F(b) = b^3/3$, and the enclosed area over $\lbrack a, b \rbrack$ is $\int_a^b x^2\,dx = \frac{b^3}{3} - \frac{a^3}{3}.$

The same reasoning gives the general power function:

Integration of the power function. For $p_i(x) = x^i$ with $i \in \mathbb{N} \cup \{0\}$: $\int_a^b x^i\,dx = \frac{b^{i+1} - a^{i+1}}{i + 1}.$

This is the "reverse" of the differentiation rule: raise the exponent by one and divide by the new exponent.

Integrability and the Definite Integral

The construction generalises to arbitrary functions through limits of Riemann sums.

Integrability and the definite integral. A function $f$ is integrable on $\lbrack a, b \rbrack$ if, for every partition into subintervals $I_{j,n} = \lbrack x_{j,n}, x_{j+1,n} \rbrack$ whose widths tend to $0$ as $n \to \infty$ and for every choice of sample points $\tilde{x}_{j,n} \in I_{j,n}$, the sums $\mathcal{F}_n = \sum_{j=0}^{n-1} f(\tilde{x}_{j,n})\,(x_{j+1,n} - x_{j,n})$ converge to one and the same limit. That limit is the definite integral $\int_a^b f(x)\,dx = \lim_{n \to \infty} \mathcal{F}_n.$

Here $f$ is the integrand, $x$ the integration variable, and $a, b$ the integration bounds. The integral sign $\int$ is a stylised sum $\sum$, reflecting that the integral is a limit of sums of rectangular areas $f(\tilde{x}_{j,n})\,(x_{j+1,n} - x_{j,n})$.

Fortunately, almost every function encountered in practice is integrable:

  • Monotone functions are always integrable.
  • Continuous functions — those whose graph can be drawn without lifting the pen — are always integrable, and this class is broad.

A function must be quite pathological to fail: the function equal to $0$ on the rationals and $1$ on the irrationals of $\lbrack 0, 1 \rbrack$ is not integrable.

Properties of the Definite Integral

For integrable functions the definite integral satisfies the following, all read off from the defining sums.

  • Additivity of intervals: $\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx.$
  • Change of bounds: $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx.$
  • Empty integral: $\int_a^a f(x)\,dx = 0.$
  • Positivity: if $f(x) \ge 0$ on $\lbrack a, b \rbrack$, then $\displaystyle\int_a^b f(x)\,dx \ge 0$.

Oriented Versus Absolute Area

Because the integrand's sign carries into the integral, the definite integral gives the oriented (signed) area, not the absolute area. For $f(x) = x$ on $\lbrack -1, 1 \rbrack$ the negative and positive parts cancel: $\int_{-1}^{1} x\,dx = 0.$

To obtain the absolute area, integrate the absolute value $\lvert f \rvert$ — equivalently, integrate $-f$ wherever $f$ is negative: $\int_{-1}^{1} \lvert x \rvert\,dx = -\int_{-1}^{0} x\,dx + \int_{0}^{1} x\,dx = \tfrac{1}{2} + \tfrac{1}{2} = 1.$

Integration Rules and Polynomials

Just as for derivatives, factor and sum rules simplify integration.

Factor and sum rules. For a real constant $\lambda$ and integrable $f$, $g$: $\int_a^b (\lambda f)(x)\,dx = \lambda \int_a^b f(x)\,dx, \qquad \int_a^b (f + g)(x)\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx.$

Combined with the power-function rule, these integrate any polynomial:

Integration of polynomials. For $p(x) = a_n x^n + \cdots + a_1 x + a_0$: $\int_a^b p(x)\,dx = a_n\,\frac{b^{n+1} - a^{n+1}}{n+1} + \cdots + a_1\,\frac{b^2 - a^2}{2} + a_0\,(b - a).$

For example $\displaystyle\int_0^2 (2x - x^2)\,dx = 2 \cdot 2 - \tfrac{8}{3} = \tfrac{4}{3}$.

Where This Leads

So far the definite integral is defined through limits of sums, which are laborious to evaluate. The next page, Antiderivatives and the Main Theorem, reveals the deep connection between integration and differentiation that turns integral evaluation into a matter of finding antiderivatives.