Interactive guide
One curve, two questions
Everything below uses the exact same simple curve, f(x) = x². A derivative answers "how steep is this curve, right at this one point?" An integral answers "how much area sits under this curve, between two points?" Both questions get answered the same way: by cheating with something easy (a straight line, a rectangle), then asking what happens as that cheat gets infinitely small.
The one trick behind both
You can't directly measure the "steepness at one exact point": a single point has no slope by itself. So instead, pick a second, nearby point, draw a straight line between them (easy, that's just slope), and then slide the second point closer and closer to the first. Whatever slope that line settles on is the derivative.
You can't directly measure "the area under a curve" either; only straight-edged shapes have easy area formulas. So instead, chop the region into thin rectangles (easy, that's just width times height), add up their areas, and make the rectangles thinner and more numerous. Whatever total that settles on is the integral.
Same move both times: replace something hard with something easy, then take it to the limit.
Same idea, read two ways
-
1
Derivative: shrink the gap
Two points get closer together. Their connecting line's slope settles on the curve's exact slope.
-
2
Integral: shrink the rectangles
More, thinner rectangles. Their total area settles on the curve's exact area.
-
3
They undo each other
The rate the area grows, at any point, is exactly the curve's height there. Part 3 below shows this directly.
Part 1: What is a derivative?
The point A sits still at x = 1. Point B slides toward it as you shrink h. Watch the line through A and B (solid) turn into the curve's actual tangent line at A (dashed).
curve secant (A to B) true tangent at A
- Slope of the line through A and B
- 3.000
- True slope of the curve at A (the derivative)
- 2.000
—
Watch it converge
| h | 1.0 | 0.5 | 0.1 | 0.01 | 0.001 |
|---|---|---|---|---|---|
| Slope of A–B |
Part 2: What is an integral?
The shaded rectangles fill the area under the curve from x = 0 to x = 2. More rectangles means thinner rectangles: watch their total area close in on the true area.
curve rectangles
- Total area of the rectangles
- 8.000
- True area under the curve (the integral)
- 2.667
—
Watch it converge
| n | 1 | 5 | 10 | 20 | 40 | 100 |
|---|---|---|---|---|---|---|
| Rectangle total |
Part 3: Why they're opposites
Slide x and watch the shaded area grow. The number that matters: how fast the area is growing right now always matches the curve's height right now: the Fundamental Theorem of Calculus, made concrete instead of memorized.
shaded region = area from 0 to x
- Area so far, A(x)
- 0.333
- How fast the area is growing right now
- 1.000
- Height of the curve right now, f(x)
- 1.000
—
A derivative shrinks a gap between two points until a line's slope becomes an exact instant. An integral grows a pile of thin rectangles until their total becomes an exact area. Part 3 is the payoff for seeing both: the area function's own growth rate, at any point, is exactly the original curve's height there. Differentiating undoes integrating, not as a rule to memorize, but as something you can watch happen.
The jargons
Derivatives and integrals turn out to be the same limiting trick run in opposite directions — shrinking a secant line into a tangent on one side, shrinking rectangles into exact area on the other — and the Fundamental Theorem is the proof that they’re inverses.
The exact slope of a curve at one point: the limit of the slope between that point and a second point, as the second point gets infinitely close.
A straight line through two points on a curve. As the two points move together, the secant line rotates into the tangent line.
The straight line that touches a curve at one point and matches its exact slope there: what the secant line becomes in the limit.
The exact area under a curve between two points: the limit of a sum of rectangle areas, as the rectangles get infinitely thin and infinitely numerous.
The total area of a specific set of rectangles approximating the area under a curve, named for the "cheat" step before taking the limit.
The area-so-far function's derivative equals the original curve: differentiation and integration are inverse operations.