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MathematicsA Level & IB

What is definite integral?

A definite integral evaluates an integral between two limits and returns a single numerical value, usually representing signed area.

Definite integral explained

You integrate as normal, then substitute the upper limit and subtract the value at the lower limit. The constant of integration cancels, which is why it never appears in a definite answer.
Definite integrals model accumulated quantities: total distance from a velocity graph, total charge from a current-time graph, or the mean value of a function over an interval.

Key formula

∫abf(x) dx=F(b)−F(a)\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)

Worked example

∫132x dx=[x2]13=9−1=8\displaystyle\int_{1}^{3} 2x\,dx = \left[x^{2}\right]_{1}^{3} = 9 - 1 = 8.

Examiner tip

Show the square-bracket line before substituting. Method marks are available there even if your arithmetic slips afterwards.

Definite integral: common questions

Can a definite integral be negative?
Yes. Regions below the x-axis contribute negative signed area, so an integral can be negative or even zero when areas cancel.
Do I need + c in a definite integral?
No — the constants cancel when you subtract, so writing + c is unnecessary although not penalised.

Related terms

Integration

Integration is the reverse of differentiation: it recovers a function from its rate of change and measures the area under a curve.

Area under a curve

The area under a curve is the region between the graph and the x-axis, found exactly by definite integration or estimated using the trapezium rule.

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

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