Definite integral
A definite integral computes the signed area between a function and the x-axis over an interval [a, b], written ∫ from a to b of f(x) dx. It evaluates to a number, found by the Fundamental Theorem of Calculus as F(b) − F(a).
A definite integral measures the accumulated total of a function over an interval. Written ∫ from a to b of f(x) dx, it gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b: regions above the axis count as positive area, regions below count as negative.
Formally, the definite integral is the limit of Riemann sums — approximating the region with thin rectangles and letting their widths shrink to zero. In practice, you rarely compute that limit directly. The Fundamental Theorem of Calculus says that if F is any antiderivative of f, then the definite integral equals F(b) − F(a). For example, ∫ from 0 to 2 of x² dx = (x³/3) evaluated from 0 to 2 = 8/3.
Unlike an indefinite integral, which is a family of functions (an antiderivative plus a constant C), a definite integral is a number. That number can represent far more than area: displacement from a velocity function (total distance instead integrates speed, |v|), total change from a rate of change, average value when divided by the interval width, and volumes or accumulated quantities in applied problems. Useful properties include splitting the interval, pulling out constants, and flipping the bounds to negate the result.
Definite integrals are a pillar of the AP Calculus AB exam — evaluating them with the Fundamental Theorem, interpreting them as net change, and estimating them from graphs or tables with Riemann sums. Engineering licensure exams such as the FE also assume fluency with definite integrals throughout their mathematics coverage.
Key takeaways
- A definite integral ∫ from a to b of f(x) dx gives the signed area under f between a and b.
- The Fundamental Theorem of Calculus evaluates it as F(b) − F(a), where F is an antiderivative of f.
- A definite integral is a number; an indefinite integral is a family of antiderivatives plus C.
- Riemann sums define the integral as a limit of rectangle approximations and are used to estimate it from data.
- AP Calculus AB and the FE exam both test setting up, evaluating, and interpreting definite integrals.