∫ₐᵇ · The Riemann Integral
Where the upper and lower sums squeeze onto one value — that value is the area.
How can you measure the area under a curve exactly? A ruler won't do it, and no memorized formula will either.
Read the roles of sample points and partitions.
Add infinitely many rectangles, yet the total stays finite — why?
Open the immersive version →A reading path containing this lesson: Make closeness precise · 3/3
Why does this question matter?
The phrase 'area under a curve' is ancient, but what that area precisely is stayed fuzzy. Rectangles and triangles have formulas, but there is no ruler for a region bounded by a bent edge.
Newton and Leibniz computed area with an antiderivative F(x). But that is a method of calculation, not a definition. It never answered 'what is area?'. No one could rigorously say why the calculation worked.
Riemann's answer is squeezing. Press down from above and brace up from below with rectangles; the instant the two sums squeeze onto the same value, that value is the area. This definition builds area with no F(x) — and at the same time reveals exactly which functions have no area at all.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
How do you measure the area under a curve exactly? You can't use a ruler, and there's no formula to memorize. Instead, squeeze it from above and below with rectangles. The arena is f(x)=x² on [0, 1]. The true area is 1/3.
Predict first — as you make the rectangles infinitely fine, what happens to the upper sum pressing down and the lower sum bracing up?
Read the idea from another angle
The integral is not the reverse of the antiderivative — it is the squeezing of area.
In high school you learn ∫ as 'differentiation backwards'. At university the order flips. First area is defined by squeezing the upper and lower sums; the link to the antiderivative is proved later, as the Fundamental Theorem. Definition first, calculation method second.
Not every function has an area.
Intuition whispers 'if there's a curve, there's an area under it'. The Dirichlet function shatters that. Only functions for which the squeeze succeeds have an area. Integrability is a qualification the function must pass.
Now read the definitions with their conditions
Translating what you squeezed by hand in the game into the language of mathematics:
Anatomy of the definition — paired with what you squeezed in stages 1 and 2
∫ₐᵇ f dx = ·
Upper and lower sums (x² on [0,1], verified)
The two numbers the stage-1 slider showed. At n=2, U=0.625 and L=0.125. As n grows, both squeeze onto 1/3.
Area defined by squeezing
The upper sum pressing down and the lower sum bracing up reach the same limit, 1/3. That single value is the definite integral.
Why the gap goes to zero
Because x² is increasing, each subinterval's (max − min) telescopes, so the gap is exactly 1/n. When it goes to zero, only one value can fit between.
General definition of the Riemann integral
Let f be bounded on a closed bounded interval [a,b]. Use the supremum and infimum on each subinterval to define U(P) and L(P). Riemann integrability means infₚ U(P)=supₚ L(P). Equivalently, as the mesh ‖P‖→0, every choice of sample points yields Riemann sums tending to the same value. The supremum and infimum need not be attained.
Functions here are bounded on a closed bounded interval. Suprema/infima need not be attained.
References
- OpenStax · Calculus Volume 2 · 3.6 Numerical Integration
Make one more distinction
The integral is the same thing as the antiderivative (finding F(x)).
In the experiment we fixed the area without ever finding F(x). The integral is defined as the squeezing of area. The link to the antiderivative is a separate theorem — the Fundamental Theorem of Calculus. Riemann defined area with no F(x) at all.
Every function has an area (an integral value).
The Dirichlet function is a counterexample: 1 on the rationals, 0 on the irrationals, so every subinterval holds both a 1 and a 0. The upper sum stays at 1 and the lower at 0, so the gap never goes to zero. It is not Riemann integrable.
Where else can this idea help?
How computers find area
The trapezoidal and Simpson's rules integrate simpler approximations to a function. Their outputs are not automatically upper and lower bounds for the true integral. Separate error bounds require assumptions such as appropriate smoothness and bounds on derivatives.
Continuous probability distributions
The probability of a continuously spread value, like height or weight, is defined as the area under a curve (the probability density function). The chance of landing in an interval is the area over it — an integral. Area is probability.
Squeezing out π
The area under the quarter circle y=√(1−x²) is π/4. Squeeze it with rectangles from above and below and π is narrowed deterministically — unlike random Monte Carlo dart-throwing, this squeeze never wobbles.
Riemann's question, Lebesgue's answer
Riemann's question — 'which functions have an area?' — led to the 20th-century Lebesgue integral. It is an extension that grants an area even to functions Riemann missed, like the Dirichlet function.
Explain it in your own words
Check the idea with practice problems (4)
Questions completed 0 / 4
1. For f(x)=x², split [0,1] into 2 equal pieces. What is the upper sum U(2)? (as a decimal, or 5/8)
2. For f(x)=x² on [0,1], why does the gap U(n)−L(n) go to zero as n grows?
3. In Riemann integration, what is 'the definition of the integral'?
4. Why is the Dirichlet function (1 on rationals, 0 on irrationals) not Riemann integrable?
What question comes next?
A reading path containing this lesson: Make closeness precise · 3/3
Previous concept
Differentiability
If magnification showed you smoothness, this is the opposite direction — cutting fine and stacking, a squeeze.
← Differentiability labLeads to next
Fundamental Theorem of Calculus
On to the astonishing theorem that squeezing area (integration) and the limit of the tangent (differentiation) are inverses of each other.
Go to the Fundamental Theorem of Calculus lab →Related lab
Integral
The intuitive version — where you first meet integration through stacking and accumulation.
Go to the Integral lab →Related lab