Key takeaway: A definite integral is a single number, the limit of Riemann sums, that gives the net signed area between a curve and the horizontal axis from one limit to another.
Grade 12 – AP Calculus ABAP Calculus AB Unit 6Calculus
Definition of Definite Integral
A definite integral measures how much builds up between two specific inputs. Regions over the horizontal axis add to the total and regions under it subtract, so the result is a net amount. That net amount can be zero even when the curve stays far from the axis, because gains and losses cancel.
Formally it is the value that Riemann sums settle on as their rectangles become infinitely thin. In practice it is computed from an antiderivative: evaluate at the upper limit, evaluate at the lower limit, and subtract. A speed graph, for example, integrates to the net distance traveled.
Formula and notation
With lower limit \(m\), upper limit \(n\), and an antiderivative \(H\) of \(h\):
\[ \int_{m}^{n} h(x)\,dx=H(n)-H(m). \]
Reversing the limits flips the sign, and \(\int_{m}^{m} h(x)\,dx=0\). The result is a number, not a function.
Key facts
- The answer is a number with units: output units times input units.
- Swapping the limits changes the sign of the result.
- Integrals over adjacent intervals add: from m to p plus from p to n equals from m to n.
- Regions below the axis reduce the total, so net area can differ from total area.
Definite Integral example
Evaluate \(\int_{1}^{3}(6x^2-4x+1)dx\).
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Antiderivative: \(F(x)=2x^3-2x^2+x\).
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Upper limit: \(F(3)=54-18+3=39\).
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Lower limit: \(F(1)=2-2+1=1\).
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Subtract: \(39-1=38\).
The definite integral equals 38.
Common mistake
Watch out
A student stops after substituting the top limit, \(F(3)=39\), and reports 39.
Why it is wrong
The evaluation formula needs both limits: \(F(3)-F(1)\). Skipping the lower-limit value treats the area as if it started at the origin, adding the extra piece from 0 to 1. Subtracting 1 leaves the correct 38.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus