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.
Key Terms
Definite Integral— ∫abf(x)dx, a number representing accumulated quantity
Riemann Sum— the approximating sum whose limit is the integral
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.
Sample point choice
Formula for xi*
Resulting Riemann sum
Left endpoint
xi* = xi−1 (left edge of i-th subinterval)
left Riemann sum — rectangle heights f(xi−1)
Right endpoint
xi* = xi (right edge of i-th subinterval)
right Riemann sum — rectangle heights f(xi)
Midpoint
xi* = (xi−1 + xi) / 2
midpoint Riemann sum — rectangle heights at the center of each subinterval
Definite Integral Notation
Notation
Definite Integral Notation
What the bounds add to the sign, which letters are disposable, and the bracket that closes every FTC computation.
The indefinite sign with bounds welded on — and the output changes species: a number, not a family. Lower limit a below, upper limit b above; sign and bounds act as one operator on the integrand.
Casesb<a is legal — reversing the bounds flips the sign, Properties of Definite Integrals below. a=b gives 0. A bound of ±∞, or an integrand blowing up inside [a,b], hands the symbol to improper integrals.
Also written∫[a,b]f — over-the-set style in analysis and measure theory, where orientation is deliberately absent.
Do not confuseThe indefinite family. No +C appears here — any antiderivative gives the same F(b)−F(a); the constant cancels.
∫01t2dt=∫01x2dx
The variable of integration is a dummy
Inside a definite integral the letter is a bound placeholder — it vanishes with the dx when the number comes out. Both spellings above name the same value, 31; so would u, τ, or anything else.
CasesThe letter turns load-bearing in accumulation functions: ∫axf(t)dtneedst inside because x is busy in the bound — ∫axf(x)dx collides the two roles. Engineering writes τ under the sign when t is real time outside it.
Also writtenξ as the standing dummy in Russian analysis texts.
Do not confuseThe indefinite integral's variable — not a dummy there: the answer F(x)+C keeps it alive.
F(x)ab · [F(x)]ab
F of x, evaluated from a to b
Shorthand for F(b)−F(a) — the closing move of every FTC computation, Computing Definite Integrals below. Upper value minus lower, always that order.
CasesBar and bracket forms are interchangeable; the bracket survives long expressions better in print. Reversing the bounds flips its sign along with the integral's.
Also writtenFab with the argument suppressed.
Do not confuseThe one-point evaluation barx=a — same glyph, different arithmetic: substitution there, subtraction of two substitutions here.
∑i=1nf(xi∗)Δx⟶∫abf(x)dx
The Riemann sum becomes the integral
The notation records its own limit, from The Riemann Sum Construction above: Σ stretches into the elongated S, Δx shrinks to dx, the sampled xi∗ relaxes into a continuous x. Leibniz designed the symbol as exactly this metamorphosis.
CasesThe star in xi∗ marks a chosen sample point — left endpoint, right endpoint, midpoint all legal; the limit forgets the choice.
Also writtenSn for the n-th Riemann sum, as above.
Same glyph elsewhereThe same superscript star marks the complex conjugatez∗ — an unrelated job in the same corner of the symbol.
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:
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).
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):
f(c)=favg
Summary: Three Lenses on the Definite Integral
The sections above introduce the definite integral from three angles: as the limit of Riemann sums (the construction), as signed area between a curve and the x-axis (the geometric meaning), and as a way to define the average value of a function on an interval (a derived use). The table below collects these three lenses together — each is a valid perspective, and each is useful in a different setting. Computing a definite integral in practice usually relies on the Fundamental Theorem of Calculus, detailed on the rules page.
Lens
Defining expression
What it gives
Geometric picture
Riemann sum (limit definition)
limn → ∞ Σi=1n f(xi*) Δx
the rigorous construction — what the integral fundamentally is
infinitely many rectangles becoming infinitesimally thin
Signed area
∫ab f(x) dx
a single number measuring net accumulation over [a, b]
area above the x-axis minus area below
Average value
(1 / (b − a)) · ∫ab f(x) dx
the mean output of f over [a, b]
height of a rectangle on [a, b] whose area equals the area under f
Definite Integrals FAQ
Why is there no + C in a definite integral?
+
Because the constant cancels. A definite integral evaluates F(b) − F(a), and using F(x) + C instead gives (F(b) + C) − (F(a) + C), where the two copies of C subtract away. Any antiderivative produces the same number, so there is nothing for + C to record. The family only survives in the indefinite integral.Read more →
Why is ∫ₐˣ f(t) dt written with t instead of x?
+
Because x is already in use as the upper bound. Inside a definite integral the variable is a dummy — ∫₀¹ t² dt and ∫₀¹ x² dx name the same number — so the letter is normally free. But an accumulation function needs x for the bound, and ∫ₐˣ f(x) dx would give one letter two jobs at once.Read more →
Does it matter whether you use left or right endpoints in a Riemann sum?
+
For any fixed number of rectangles, yes — left endpoints, right endpoints, and midpoints generally give three different numbers. In the limit, no: as the rectangles become infinitely thin all three converge to the same definite integral, provided f is integrable. The star in x* marks the sample point precisely because the choice is free.Read more →
How do you find the total area between a curve and the x-axis?
+
Integrate the absolute value: total area = ∫ₐᵇ |f(x)| dx. The plain definite integral gives signed area, counting regions below the axis as negative, so it can come out smaller than the true area or even zero when the parts cancel. In practice, find where f crosses the axis, integrate each piece separately, and add the absolute values.Read more →
What is the Mean Value Theorem for Integrals?
+
It guarantees that a continuous function actually attains its own average value somewhere on the interval: there is at least one c in (a, b) with f(c) = f_avg, where f_avg = (1/(b − a))∫ₐᵇ f(x) dx. Geometrically, the rectangle of height f(c) over [a, b] has exactly the area under the curve.Read more →