Definite Integral¶

Definition¶

Let $a,b\in\mathbb{R}$ with $a<b\text{.}$ Suppose $f\colon I\to\mathbb{R}$ is a Riemann integrable function on $I=[a,b]\text{.}$ The following expression

is referred to as a definite integral, which is a real number, not a function of $x\text{.}$

The symbol $\int$ is a stylized "S" for "sum", while $\mathrm{d}x$ records the variable and the limiting width:

Here $x_i^*$ is any point in $[a+(i-1)(b-a)/n,\,a+i(b-a)/n]$$a)/n]$, the $i$th subinterval of the regular subdivision.

The function $f$ is the integrand; $a$ and $b$ are the lower and upper limits of integration. The letter $x$ is a dummy variable, so $\int_a^b f(x)\,\mathrm{d}x=\int_a^b f(t)\,\mathrm{d}t\text{.}$

Interpretation¶

The definite integral $\int_a^b f(x)\,\mathrm{d}x$ is a single number that adds up the values of $f$ over $[a,b]$, each weighted by the length over which it holds: it is the limit of the sums $\displaystyle \sum f(x_i^*)\frac{b-a}{n}$ in the Definition. Geometrically it is a net area.

The figure shows $f(x)=\sin x$ from $a$ to a point $b$ beyond $\pi$. Up to $\pi$ the graph lies above the axis, and the blue area counts positively; between $\pi$ and $b$ it lies below the axis, and the red area counts negatively. The integral is blue minus red, so it is smaller than the total shaded area.

Each part of the notation has a role. The integrand $f$ supplies the heights, $a$ and $b$ fix the interval, and $\mathrm{d}x$ names the variable of integration and stands for the shrinking widths $\frac{b-a}{n}$. The variable is a dummy, so the value does not depend on $x$: $\int_a^b f(x)\,\mathrm{d}x$ is a number, while $\int_a^x f(t)\,\mathrm{d}t$ is a function of its upper limit.

The fundamental theorem of calculus evaluates a definite integral as $F(b)-F(a)$ for an antiderivative $F$, so in practice it is rarely computed from sums.

A common slip is to read the integral as the area itself. Signed area and area differ when $f$ changes sign: $\int_{-1}^1 x\,\mathrm{d}x=0$$0$, although the region between the graph and the axis has area $\int_{-1}^1|x|\,\mathrm{d}x=1$$1$.

Later, improper integrals extend the definite integral to unbounded intervals and integrands.

Key Results¶

• If $f$ is integrable on $[a,b]$ and $a<c<b\text{,}$ then $\displaystyle\int_a^b f=\int_a^c f+\int_c^b f\text{.}$
• The value of a definite integral does not depend on any partition: refining a partition can only decrease upper sums and increase lower sums, and for an integrable function both approach the same number.
• A definite integral measures signed area: regions above the $x$-axis contribute positively and regions below contribute negatively. When $f\geq0\text{,}$ it agrees with the ordinary area under the graph.
• By convention, $\displaystyle\int_b^a f=-\int_a^b f$ for $a<b$ and $\displaystyle\int_a^a f=0\text{;}$ hence reversing the limits always changes the sign: $\displaystyle\int_u^v f=-\int_v^u f\text{.}$
• Varying an endpoint produces the functions $\displaystyle F(x)=\int_a^x f(t)\,\mathrm{d}t$ and $\displaystyle G(x)=\int_x^b f(t)\,\mathrm{d}t$ on $[a,b]\text{,}$ with $F(a)=0$ and $G(b)=0\text{.}$ Write $\int_a^x f(t)\,\mathrm{d}t\text{,}$ never $\int_a^x f(x)\,\mathrm{d}x\text{,}$ since the latter uses $x$ in two different roles.

• A definite integral does not depend on the name of the integration variable, i.e.,

  $$\int_a^b f(x)\,\mathrm{d}x=\int_a^b f(y)\,\mathrm{d}y.$$

Examples¶

• $\displaystyle\int_0^1 x\,\mathrm{d}x=\frac{1}{2}\text{.}$
• $\displaystyle\int_{-1}^1 x\,\mathrm{d}x=0\text{.}$

Prerequisites¶

• Riemann Integral

Related Concepts¶

• Indefinite Integral
• Differential
• Fundamental Theorem of Calculus Part 1
• Riemann Sum
• Integrand
• LOTUS

Sources¶

• Math I: Calculus · Lecture 11

Symbols: element of, infinity, integral, pi, real numbers, sum

Last updated: 20 October 2026 03:02