How can you measure the area under a curve exactly? A ruler won't do it, and no memorized formula will either.
The integral is not the reverse of the antiderivative. When the upper sum pressing down and the lower sum bracing up squeeze onto the same value, the value caught between them is the area.
Experiment
Hands-on experiment
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 more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
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.
Insight
Insights from the video
“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.
Misconception
Common misconceptions
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.
Formula
Writing it as math
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
As the partition P is made infinitely fine, if the upper and lower sums become equal, that value is the integral. If they differ, the integral is undefined.
In Real Life
Where you meet it in real life
How computers find area
Even for integrals no one can do by hand, computers find the area by squeezing it. The trapezoidal and Simpson's rules are the practical version of this upper/lower squeeze. Numerical integration is Riemann's definition turned into code.
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.
Practice
Practice — conquer the types
For f(x)=x², split [0,1] into 2 equal pieces. What is the upper sum U(2)? (as a decimal, or 5/8)
For f(x)=x² on [0,1], why does the gap U(n)−L(n) go to zero as n grows?
In Riemann integration, what is 'the definition of the integral'?
Why is the Dirichlet function (1 on rationals, 0 on irrationals) not Riemann integrable?
Watch
Related video
Connection
Concepts connect
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
Labs worth exploring together
Related lab
Integral
The intuitive version — where you first meet integration through stacking and accumulation.
Go to the Integral lab →Related lab