What is a Riemann sum?

A Riemann sum approximates a definite integral by partitioning the interval into subintervals, forming rectangles with heights f(x*) and width Δx, then summing their areas. As the number of rectangles approaches infinity, the Riemann sum approaches the definite integral.

What does definite integral notation mean?

In ∫ₐᵇ f(x) dx, the lower limit a and upper limit b bound the interval, f(x) is the integrand being accumulated, and dx indicates the variable of integration. The result is a number, not a function. The variable x is a dummy variable that disappears after integration.

What is signed area in integration?

The definite integral computes area with sign: regions above the x-axis contribute positive area, regions below contribute negative area. The integral sums these signed contributions, so it can be zero (when areas cancel) or negative (when more area lies below the axis).

What are the properties of definite integrals?

Key properties include: additivity over intervals (∫ₐᵇ + ∫ᵇᶜ = ∫ₐᶜ), reversing limits negates the integral (∫ₐᵇ = −∫ᵇₐ), zero-width intervals give zero (∫ₐᵃ = 0), and comparison (if f ≤ g then ∫f ≤ ∫g).

What is the linearity property of integrals?

Definite integrals respect addition and scalar multiplication: ∫[f + g] = ∫f + ∫g (sum rule) and ∫c·f = c·∫f (constant multiple rule). This allows complex integrands to be broken into simpler pieces, integrated separately, then combined.

How do you compute a definite integral?

The Fundamental Theorem of Calculus provides the method: find an antiderivative F where F'(x) = f(x), then evaluate ∫ₐᵇ f(x) dx = F(b) − F(a). This transforms integration from a limiting process into finding antiderivatives and evaluating at endpoints.

Definite Integrals

Q: What is the average value of a function?

The average value of f on [a,b] is f_avg = (1/(b−a))∫ₐᵇ f(x) dx. Geometrically, it's the height of a rectangle with base [a,b] whose area equals the area under the curve. The Mean Value Theorem guarantees f attains this average at some point c in (a,b).

The Riemann Sum Construction

Notation and Meaning

Signed Area Interpretation

Properties of Definite Integrals

Linearity

Computing Definite Integrals

Average Value of a Function

Summary: Three Lenses on the Definite Integral

From Sums to Areas

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

\int_a^b f(x)\, dx

specifies the integrand

f(x)

, the variable of integration

x

, and the bounds from

a

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

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—

\int_a^b f(x)\,dx

, a number representing accumulated quantity

Riemann Sum— the approximating sum whose limit is the integral

Bounds of Integration— where accumulation starts (

a

) and ends (

b

)

Signed Area— positive above the axis, negative below

Average Value of a Function—

\frac{1}{b-a}\int_a^b f(x)\,dx

Integrand— the function being accumulated

See All Calculus Definitions →

Notation and Meaning

The definite integral

\int_a^b f(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

\int_0^1 t^2\, dt \qquad \int_0^1 u^2\, du \qquad \int_0^1 x^2\, dx

all represent the same value:

1/3

.

See All Calculus Symbols and Notations →

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:

\int_a^b f(x)\, dx = (\text{area above}) - (\text{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 Unsigned Area

\text{Total area} = \int_a^b |f(x)|\, dx

Learn more about this formula: Total Unsigned Area →

Browse all integral formulas →

Properties of Definite Integrals

Definite integrals satisfy several fundamental properties.

Additivity over intervals:

\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx

Reversing limits negates the integral:

\int_a^b f(x)\, dx = -\int_b^a f(x)\, dx

Zero-width interval:

\int_a^a f(x)\, dx = 0

Comparison: If

f(x) \leq g(x)

[a, b]

, then

\int_a^b f(x)\, dx \leq \int_a^b g(x)\, dx

Linearity

Definite integrals respect addition and scalar multiplication.

Sum rule:

\int_a^b [f(x) + g(x)]\, dx = \int_a^b f(x)\, dx + \int_a^b g(x)\, dx

Constant multiple rule:

\int_a^b c \cdot f(x)\, dx = c \int_a^b f(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:

\int_a^b f(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) \Big|_a^b

[F(x)]_a^b

denotes the evaluation

F(b) - F(a)

Average Value of a Function

The average value of

f

on the interval

[a, b]

is:

Average Value of a Function

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\, dx

Learn more about this formula: Average Value of a Function →

Browse all integral formulas →

This generalizes the familiar average of discrete values. The integral computes the total, and division by the interval length yields the mean.

Geometrically,

f_{\text{avg}}

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

(a, b)

f(c) = f_{\text{avg}}

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)	lim_{n → ∞} Σ_i=1ⁿ f(x_i^*) Δx	the rigorous construction — what the integral fundamentally is	infinitely many rectangles becoming infinitesimally thin
Signed area	∫_a^b 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)) · ∫_a^b 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