Bounds of Integration — Definition, Formula & Examples

Bounds of Integration
Limits of Integration

For the definite integral Definite integral notation: integral from a to b of f(x)dx, where a and b are the lower and upper bounds of integration. , the bounds (or limits) of integration are a and b.

Key Formula

\int_a^b f(x)\,dx

Where:

$a$ = Lower bound of integration (starting value)
$b$ = Upper bound of integration (ending value)
$f(x)$ = The function being integrated (the integrand)

Worked Example

Problem: Identify the bounds of integration and evaluate the definite integral

\int_1^4 2x\,dx

Step 1: Identify the bounds. The lower bound is 1 and the upper bound is 4.

a = 1, \quad b = 4

Step 2: Find the antiderivative of

2x

F(x) = x^2

Step 3: Evaluate the antiderivative at the upper bound and subtract its value at the lower bound.

F(4) - F(1) = 4^2 - 1^2 = 16 - 1 = 15

Answer: The bounds of integration are 1 (lower) and 4 (upper), and the definite integral equals 15.

Why It Matters

Bounds of integration determine exactly which portion of a function you are accumulating. Changing the bounds changes the result entirely — for instance, the area under a curve from 0 to 2 is generally different from the area from 0 to 5. In applications, the bounds often represent physical constraints such as a time interval, a range of positions, or the start and end of a process.

Common Mistakes

Mistake: Subtracting in the wrong order, computing

F(a) - F(b)

instead of

F(b) - F(a)

Correction: Always evaluate as upper bound minus lower bound:

F(b) - F(a)

. Reversing the bounds flips the sign of the integral.

Related Terms

Definite Integral — The integral type that uses bounds of integration
Indefinite Integral — An integral without bounds, yielding a family of functions
Fundamental Theorem of Calculus — Connects antiderivatives to evaluation at the bounds
Integrand — The function being integrated between the bounds