Definite Integration

This lesson covers definite integration — evaluating integrals between given limits to find exact numerical values — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to understand how to evaluate definite integrals, find areas under curves, and handle areas below the x-axis.

---

What Is a Definite Integral?

A definite integral has upper and lower limits of integration. It produces a numerical value rather than a function.

The notation is:

∫ from a to b of f(x) dx

where a is the lower limit and b is the upper limit.

Evaluating a Definite Integral

To evaluate a definite integral:

1. Find the indefinite integral (antiderivative) of f(x) — call it F(x).
2. Evaluate F(b) - F(a).

This is written using square bracket notation:

∫ from a to b of f(x) dx = [F(x)] from a to b = F(b) - F(a)

Note: There is no constant of integration c when evaluating definite integrals. The constants would cancel: [F(x) + c] from a to b = (F(b) + c) - (F(a) + c) = F(b) - F(a).

---

Worked Examples

Example 1: Evaluate ∫ from 1 to 3 of 2x dx