Integral Sign — Definition, Formula & Examples

The integral sign (∫) is the elongated S-shaped symbol used in calculus to indicate that you are finding the integral of a function. It tells you to sum up infinitely many infinitely small pieces, such as the area under a curve.

The symbol ∫, introduced by Leibniz in 1675, denotes the operation of integration. In a definite integral $\int_a^b f(x)\,dx$ , it specifies that the function $f(x)$ is to be integrated with respect to $x$ over the interval from $a$ to $b$ . The symbol derives from a stylized Latin letter "s" for "summa" (sum).

Key Formula

\int_a^b f(x)\,dx

Where:

$\int$ = The integral sign, indicating the operation of integration
$a$ = Lower limit of integration
$b$ = Upper limit of integration
$f(x)$ = The integrand — the function being integrated
$dx$ = Indicates that integration is performed with respect to x

How It Works

The integral sign never appears alone — it is part of a larger expression. Immediately after ∫ you write the function to be integrated (the integrand), and at the end you write

dx

to show which variable you are integrating with respect to. For a definite integral, a lower limit is written at the bottom of the sign and an upper limit at the top, like

\int_0^3

. For an indefinite integral (no limits), the result includes a constant of integration

C

. Reading

\int_1^4 2x\,dx

aloud, you would say "the integral from 1 to 4 of 2x with respect to x."

Worked Example

Problem: Evaluate the definite integral

\int_0^3 2x\,dx

Read the notation: The integral sign with limits 0 and 3 tells you to find the accumulated area under

2x

from

x=0

x=3

\int_0^3 2x\,dx

Find the antiderivative: The antiderivative of

2x

x^2

\int 2x\,dx = x^2 + C

Apply the limits: Substitute the upper and lower limits and subtract.

\left[x^2\right]_0^3 = 3^2 - 0^2 = 9

Answer: The value of the integral is

9

, which represents the area under the line

y=2x

from

x=0

x=3

Why It Matters

You will encounter the integral sign throughout AP Calculus, physics, and engineering courses whenever quantities like area, volume, work, or accumulated change are computed. Recognizing each part of the notation — limits, integrand, and differential — is essential before you can set up or solve any integral.

Common Mistakes

Mistake: Forgetting the

dx

at the end of the integral expression.

Correction: The

dx

is not optional decoration. It specifies the variable of integration and closes the integral expression. Without it, the notation is incomplete and ambiguous.