The definite integral

The integral of 4 − x² from 0 to 3 is 3: the 16/3 above the axis minus the 7/3 that dips below it.

The limit of the sums

The definite integral of f from a to b is the limit of its Riemann sums as the number of pieces grows without bound: ∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i*)Δx, provided the limit exists and does not depend on which sample points are chosen. When it exists, f is integrable on [a, b]. OpenStax notes that every continuous function on a closed interval is integrable, so for the functions in this course the limit always exists.

The notation carries the picture. The long S, ∫, is a stretched letter for sum, introduced by Leibniz. The numbers a and b are the limits of integration, f(x) is the integrand, and dx stands for the width of a vanishingly thin rectangle and tells you which variable is being summed over. The letter x itself is a dummy variable: ∫₀² x² dx and ∫₀² t² dt are the same number, 8/3.

Net signed area

If f dips below the x-axis, the rectangles there have negative heights, so they subtract. The integral measures net signed area: area above the axis minus area below. For f(x) = 4 − x² on [0, 3], the curve is above the axis on [0, 2], with area 16/3, and below on [2, 3], with area 7/3. So ∫₀³ (4 − x²) dx = 16/3 − 7/3 = 3. If you want the total area regardless of sign, integrate |f| instead, which here gives 16/3 + 7/3 = 23/3. Which one a problem wants depends on the question: a net change, or an amount of paint.

4 − x² on [0, 3] with 12 midpoint rectangles. Those below the axis count as negative; the net is 16/3 − 7/3 = 3.

Signed area also explains a useful symmetry. If f is odd, meaning f(−x) = −f(x) as for x³ or sin x, then the area on the left of the y-axis is the mirror image of the area on the right, flipped below the axis, so ∫_−a^a f(x) dx = 0. For an even function such as x², the two halves are equal, and the integral over [−a, a] is twice the integral over [0, a]. Spotting symmetry first can save a whole calculation.

Rules and geometry

A few properties follow from the definition. An integral over a single point is zero: ∫_a^a f dx = 0. Swapping the limits changes the sign: ∫_b^a f dx = −∫_a^b f dx. Integrals over neighbouring intervals add: ∫_a^b + ∫_b^c = ∫_a^c. Constants factor out, and the integral of a sum is the sum of the integrals. When the region is a familiar shape, geometry gives the answer directly: ∫₀³ 2x dx is a triangle with base 3 and height 6, area 9; ∫₀⁴ 3 dx is a 4 by 3 rectangle, 12; and ∫₋₁¹ √(1 − x²) dx is half a unit disk, π/2.

Drag to turn. Scroll or pinch to zoom.

4 − x² from 0 to b. Rectangles below the axis turn red and count as negative. Move b from 2 to 3 and watch the exact integral fall from 16/3 to 3.

∫₀³ 2x dx (use a triangle)?
∫₀⁴ 3 dx = ?
∫₋₁¹ √(1 − x²) dx to 4 decimals?
∫₅⁵ x³ dx = ?
∫₀² f dx = 5. What is ∫₂⁰ f dx?
∫₀² f dx = 5 and ∫₂⁵ f dx = −1. ∫₀⁵ f dx = ?
Same f: ∫₀² 3f(x) dx = ?
∫₀³ (4 − x²) dx = ?
The total (unsigned) area between 4 − x² and the axis on [0, 3] is:
- 23/3
- 3
- 16/3
- 7/3
Which functions are guaranteed to be integrable on [a, b]?
- continuous ones
- only polynomials
- only positive ones
- only increasing ones

Key points

∫_a^b f(x) dx = lim_n→∞ Σ f(x_i*)Δx.
It is net signed area: above the axis minus below.
∫_a^a = 0, swapping limits flips the sign, adjacent intervals add.
Familiar shapes can be integrated with geometry.

Sources for this course (6)

OpenStax, Calculus Volume 1 , chapter 5 Integration: 5.1 Approximating Areas (Archimedes and the method of exhaustion; sigma notation with Σ 3^i for i = 1 to 5 = 363; left and right endpoint approximations L4 = 1.75 and R4 = 3.75 for x² on [0, 2], L6 = 3.4375 and R6 = 5.6875 for x²/2 on [0, 3]; (x − 1)³ + 4 on [0, 2] approaching 8; Riemann sums named for the 19th-century mathematician Bernhard Riemann; upper and lower sums: openstax.org/books/calculus-volume-1/pages/5-1-approximating-areas)
5.2 The Definite Integral (limit of Riemann sums, integrability of continuous functions, net signed area, properties: .../pages/5-2-the-definite-integral)
5.3 The Fundamental Theorem of Calculus (Newton and Leibniz, among others, late 1600s and early 1700s; mean value theorem for integrals; parts 1 and 2: .../pages/5-3-the-fundamental-theorem-of-calculus)
5.4 Integration Formulas and the Net Change Theorem (.../pages/5-4-integration-formulas-and-the-net-change-theorem)
4.10 Antiderivatives (.../pages/4-10-antiderivatives)
MacTutor, University of St Andrews: Archimedes (287 BC to 212 BC; area of a parabolic segment; π bounded with 96-sided inscribed and circumscribed polygons: mathshistory.st-andrews.ac.uk/Biographies/Archimedes) and Leibniz (first used the ∫ f(x) dx notation in a manuscript of 21 November 1675: mathshistory.st-andrews.ac.uk/Biographies/Leibniz).

Checked: 2026-10-17