The definite integral answers a concrete question: what is the total accumulated quantity between two points? Geometrically, this corresponds to the area between a curve and the horizontal axis—but with a crucial refinement. Regions below the axis contribute negative area, making the integral a signed quantity.
The notation
∫abf(x)dx
specifies the integrand f(x), the variable of integration x, and the bounds from a to b. The result is a single number, not a function. This number represents net accumulation: the sum of infinitely many infinitesimal contributions f(x)dx as x traverses the interval.
Riemann sums provide the rigorous foundation. Approximate the region with rectangles, compute their total area, and take the limit as the rectangles become infinitely thin. What emerges is not an approximation but an exact value—the definite integral.
The Riemann Sum Construction
The definite integral arises as a limit of approximating sums.
Partition the interval [a,b] into n subintervals of width Δx=(b−a)/n. In each subinterval, choose a sample point xi∗ and form the rectangle with height f(xi∗) and width Δx. The total area of these rectangles is the Riemann sum:
Sn=i=1∑nf(xi∗)Δx
As n→∞ and the rectangles become infinitely thin, the Riemann sum approaches the definite integral:
∫abf(x)dx=n→∞limi=1∑nf(xi∗)Δx
The choice of sample points—left endpoints, right endpoints, midpoints—affects individual Riemann sums but not the limit, provided f is integrable.
Notation and Meaning
The definite integral
∫abf(x)dx
consists of several components. The lower limit a and upper limit b bound the interval of integration. The integrand f(x) specifies what is being accumulated. The differential dx indicates the variable of integration.
The result is a number, not a function. The variable x is a dummy variable—a placeholder that disappears after integration. The expressions
∫01t2dt∫01u2du∫01x2dx
all represent the same value: 1/3.
Signed Area Interpretation
The definite integral computes area with sign.
Where f(x)>0, the region between the curve and the x-axis lies above the axis and contributes positive area. Where f(x)<0, the region lies below the axis and contributes negative area.
The integral sums these signed contributions:
∫abf(x)dx=(area above)−(area below)
This means the integral can be zero even when substantial area exists—positive and negative regions may cancel. It can also be negative when the curve lies predominantly below the axis.
To find total unsigned area, integrate the absolute value:
Total area=∫ab∣f(x)∣dx
Properties of Definite Integrals
Definite integrals satisfy several fundamental properties.
Additivity over intervals:
∫abf(x)dx+∫bcf(x)dx=∫acf(x)dx
Reversing limits negates the integral:
∫abf(x)dx=−∫baf(x)dx
Zero-width interval:
∫aaf(x)dx=0
Comparison: If f(x)≤g(x) on [a,b], then
∫abf(x)dx≤∫abg(x)dx
Linearity
Definite integrals respect addition and scalar multiplication.
Sum rule:
∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx
Constant multiple rule:
∫abc⋅f(x)dx=c∫abf(x)dx
These rules allow complex integrands to be broken into simpler pieces, each integrated separately and then combined.
Computing Definite Integrals
Direct computation via Riemann sums is tedious. The Fundamental Theorem of Calculus provides the shortcut.
If F is any antiderivative of f—meaning F′(x)=f(x)—then:
∫abf(x)dx=F(b)−F(a)
This result, detailed on the rules page, transforms integration from a limiting process into a two-step procedure: find an antiderivative, then evaluate at the endpoints.
The notation F(x)ab or [F(x)]ab denotes the evaluation F(b)−F(a).
Average Value of a Function
The average value of f on the interval [a,b] is:
favg=b−a1∫abf(x)dx
This generalizes the familiar average of discrete values. The integral computes the total, and division by the interval length yields the mean.
Geometrically, favg is the height of a rectangle with base [a,b] whose area equals the area under the curve. The Mean Value Theorem for Integrals guarantees that a continuous function actually attains this average value at some point c in (a,b):