The question an integral answers: how much area is under this curve?
A definite integral, written ∫ₐᵇ f(x) dx, answers a specific geometric question: what is the net signed area between the graph of f and the x-axis, from x = a to x = b? "Net signed" means that area where f(x) is positive (the curve is above the x-axis) counts positively, while area where f(x) is negative (the curve is below the x-axis) counts negatively and subtracts from the total.
This is a different question from finding a derivative, which measures a rate of change at a single point. An integral instead accumulates a quantity — area — across an entire interval, which is why integrals appear naturally whenever a total is being built up from a continuously varying rate, such as total distance traveled from a varying speed, or total accumulated quantity from a varying rate of production.
Riemann sums: approximating area with rectangles
A Riemann sum approximates the area under a curve by slicing the interval [a, b] into n smaller subintervals, drawing a rectangle over each one whose height matches the function's value at some point within that subinterval, and adding up the areas of all the rectangles. The three common variants — left, right, and midpoint Riemann sums — differ only in which point within each subinterval is used to set the rectangle's height: the left endpoint, the right endpoint, or the midpoint.
As the number of subintervals n increases, each rectangle becomes narrower and the jagged, rectangular approximation hugs the true curve more closely, so the Riemann sum's total approaches the exact area. The definite integral is formally defined as the limit of a Riemann sum as n approaches infinity (equivalently, as the width of each rectangle approaches zero) — this limit is what makes the integral a precisely defined number rather than just an approximation.
| Subintervals (n) | Left Riemann sum | Midpoint Riemann sum |
|---|---|---|
| 4 | 0.218750 | 0.328125 |
| 8 | 0.273438 | 0.332031 |
| 16 | 0.302734 | 0.333008 |
| 100 | 0.328350 | 0.333325 |
From Riemann sums to Simpson's rule: the method this calculator uses
The table above shows both a left Riemann sum and a midpoint Riemann sum for f(x) = x² on [0, 1] converging toward the exact value, 1/3 ≈ 0.33333333, as the number of subintervals n grows — the midpoint version converges noticeably faster than the left-endpoint version for this function, illustrating that not all Riemann sum variants converge at the same rate.
This site's integral calculator does not use rectangles at all; it uses adaptive composite Simpson's rule, a numerical-integration method that fits a parabolic arc across each pair of adjacent subintervals (rather than a flat rectangle top) and sums the areas under those parabolas. Because a parabola can follow a curve's bend far more closely than a flat rectangle top can, Simpson's rule reaches a given accuracy with far fewer subintervals than a Riemann sum needs — the calculator doubles the number of subintervals repeatedly until successive estimates agree to a tight numerical tolerance, then reports that converged value.
The Fundamental Theorem of Calculus, stated accessibly
The Fundamental Theorem of Calculus connects two calculus operations that seem, at first, to measure completely different things — the derivative (instantaneous rate of change) and the definite integral (accumulated area) — and shows they are inverse operations of each other. Informally: if F is any antiderivative of f (meaning F'(x) = f(x)), then the definite integral of f from a to b equals F(b) − F(a), the change in F's value across the interval.
This theorem is what makes exact, symbolic evaluation of many integrals possible without ever summing rectangles or parabolas by hand: finding one antiderivative formula lets any definite integral of that function be evaluated by simple subtraction. This site's integral calculator does not use this route — it computes the integral numerically with Simpson's rule directly, without first finding or needing an antiderivative formula, which is part of why it can integrate functions that do not have a simple closed-form antiderivative.
Worked example: ∫₀¹ x² dx = 1/3
The function f(x) = x² has the antiderivative F(x) = x³/3, since differentiating x³/3 gives back x² by the power rule. Applying the Fundamental Theorem of Calculus: ∫₀¹ x² dx = F(1) − F(0) = (1³/3) − (0³/3) = 1/3 − 0 = 1/3 ≈ 0.33333333.
This exact symbolic answer matches both the Riemann sum table above, which approaches 1/3 as the number of rectangles grows, and the numerical Simpson's-rule result this calculator would return for the same bounds — three different routes (symbolic antidifferentiation, the limit of Riemann sums, and adaptive numerical quadrature) converging on the identical value, 1/3, because they are all computing the same net signed area.
Preguntas frecuentes
What does a definite integral represent?
A definite integral, ∫ₐᵇ f(x) dx, represents the net signed area between a function's graph and the x-axis from x = a to x = b. Area above the x-axis (where f is positive) counts positively, and area below the x-axis (where f is negative) counts negatively and subtracts from the total.
What is a Riemann sum?
A Riemann sum approximates the area under a curve by dividing an interval into smaller subintervals, drawing a rectangle over each one, and adding up the rectangles' areas. As the number of subintervals grows and each rectangle narrows, the Riemann sum's total approaches the exact area — the definite integral is formally defined as the limit of this process.
How is Simpson's rule different from a Riemann sum?
A Riemann sum approximates area using flat-topped rectangles, while Simpson's rule fits a parabolic arc across each pair of subintervals and sums the areas under those parabolas instead. Because a parabola can follow a curve's bend more closely than a flat rectangle top, Simpson's rule generally reaches high accuracy with far fewer subintervals than a comparable Riemann sum.
What does the Fundamental Theorem of Calculus say, in plain English?
It says that finding a definite integral (accumulated area) and finding a derivative (instantaneous rate of change) are inverse operations: if F is an antiderivative of f, the definite integral of f from a to b equals F(b) − F(a). This lets many integrals be evaluated by simple subtraction once an antiderivative formula is known, instead of summing infinitely many rectangles.
What is ∫₀¹ x² dx and how is it calculated?
It equals exactly 1/3 (approximately 0.33333333). Using the Fundamental Theorem of Calculus with the antiderivative F(x) = x³/3: ∫₀¹ x² dx = F(1) − F(0) = 1/3 − 0 = 1/3. Numerical methods such as Riemann sums and Simpson's rule converge to this same value as their subdivisions increase.
Referencias
- Stewart J. Calculus. Cengage Learning (Riemann sums, the definite integral, the Fundamental Theorem of Calculus, and Simpson's rule).
- Burden RL, Faires JD. Numerical Analysis. Cengage Learning (composite Simpson's rule and its error bounds).
- Atkinson KE. An Introduction to Numerical Analysis. Wiley (numerical integration methods).
- NIST Digital Library of Mathematical Functions (DLMF), Chapter 3: Numerical Methods — quadrature. dlmf.nist.gov.