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Bounds of Integration — Definition, Formula & Examples

Bounds of Integration
Limits of Integration

For the definite integral , the bounds (or limits) of integration are a and b.

Key Formula

∫abf(x) dx\int_a^b f(x)\,dx
Where:
  • aa = Lower bound of integration (starting value)
  • bb = Upper bound of integration (ending value)
  • f(x)f(x) = The function being integrated (the integrand)

Worked Example

Problem: Identify the bounds of integration and evaluate the definite integral ∫142x dx\int_1^4 2x\,dx.
Step 1: Identify the bounds. The lower bound is 1 and the upper bound is 4.
a=1,b=4a = 1, \quad b = 4
Step 2: Find the antiderivative of 2x2x.
F(x)=x2F(x) = x^2
Step 3: Evaluate the antiderivative at the upper bound and subtract its value at the lower bound.
F(4)−F(1)=42−12=16−1=15F(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)F(a) - F(b) instead of F(b)−F(a)F(b) - F(a).
Correction: Always evaluate as upper bound minus lower bound: F(b)−F(a)F(b) - F(a). Reversing the bounds flips the sign of the integral.

Related Terms