Definite Integral
In short: Definite integral is the limit of Riemann sums over an interval [a, b], giving the net signed area between a function’s graph and the x-axis from a to b.
AP Calculus ABCalculusAP Calculus AB Unit 6
What is Definite Integral?
A definite integral returns one number for a function over a chosen interval: the total accumulation from \(x = a\) to \(x = b\). Picture slicing the region under the graph into thinner and thinner strips and adding their areas.
Written \(\int_a^b f(x)\,dx\), it is the limit of Riemann sums as the strips get infinitely thin. Parts of the graph above the x-axis add, and parts below subtract, so the result is a net area. When \(f\) is a rate of change, the number is the net change in the quantity.
How to spot it
- Numbers sit at the top and bottom of the integral sign, as in the integral from 1 to 4.
- A problem asks for the area under a curve between two vertical lines.
- A rate function is given and you must find the net change between two times.
- A graph of a rate crosses the axis and the question wants signed area, not total area.
The method
- Find an antiderivative \(F(x)\) of the function, ignoring \(+C\).
- Evaluate it at the top limit and at the bottom limit.
- Subtract: \(\int_a^b f(x)\,dx = F(b) – F(a)\).
- Interpret the sign and units: a negative value means more area lies below the axis than above.
Definite Integral: solved example
Problem
Evaluate \(\int_{-1}^{2} (x^2 + 1)\,dx\).
Solution
An antiderivative is \(F(x) = \frac{x^3}{3} + x\).
\(F(2) = \frac{8}{3} + 2 = \frac{14}{3}\) and \(F(-1) = -\frac{1}{3} – 1 = -\frac{4}{3}\).
Subtract: \(\frac{14}{3} – \left(-\frac{4}{3}\right) = \frac{18}{3} = \) 6.
Try it
Question 1
Evaluate \(\int_0^{\pi} \sin x\,dx\).
Show the answer
\(\left[-\cos x\right]_0^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2\).
Question 2
What is \(\int_4^4 (x^3 – 9x)\,dx\), and why?
Show the answer
It equals 0: the interval has zero width, so no area accumulates, whatever the function is.
Definite Integral vs. Antiderivative
An antiderivative is a function, found up to a constant. The definite integral is a number that uses an antiderivative as a tool, evaluated at two endpoints and subtracted. The constant \(C\) cancels in the subtraction, which is why it is never written here.
Keep going
Lessons
Worksheets and quizzes
- Riemann Sums and the Definite Integral – Worksheet
- Riemann Sums and the Definite Integral – Quiz
- Fundamental Theorem of Calculus and Antiderivatives – Worksheet
- Fundamental Theorem of Calculus and Antiderivatives – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus