Why area equals slope

What you’ll learn

The last lesson handed you the fundamental theorem as a tool; it worked, but it should bother you a little. Why would the area under a curve equal a difference of values of some related function? Area and slope look completely unrelated. The single capability this lesson builds: explain geometrically why integration and differentiation are inverse operations, by proving the fundamental theorem from one picture.

Define the area function A(x) = ∫_a^x f(t) dt, the area under f accumulated from a fixed start a to a movable right end x, and ask how fast it grows. Slide the right end out by a tiny dx: the new area is a thin sliver of width dx and height f(x), so A(x + dx) - A(x) ≈ f(x)·dx. Divide by dx, take the limit, and you get A'(x) = f(x): the derivative of the area function is the original curve. That single fact is the fundamental theorem, because it makes A an antiderivative of f, so F(b) - F(a) = ∫_a^b f(x) dx. You will see it on familiar curves (x², e^x, sin x, the circle’s 2πr), in everyday terms (a bucket filling at the tap’s flow rate; a car’s odometer and speedometer), and as the PDF-and-CDF pairing F'(x) = f(x) in machine learning.

Where this fits

This is lesson 11 of Phase 3 (Integration and approximation). It is the geometric “why” behind lesson 10’s fundamental theorem, and it bookends the track’s opening: lesson 1 noticed, on a circle, that the rate of the accumulated area is the circumference being accumulated; this lesson proves that for every curve as A'(x) = f(x). With both halves of calculus now built and bound together, the final two lessons go deeper into rates, higher-order derivatives (lesson 12) and Taylor series (lesson 13).

Before you start

Prerequisite (within this track): lesson 10, Integration and the fundamental theorem, since this lesson proves the theorem that lesson stated. You also lean on the derivative-as-a-limit idea (lesson 2), because the sliver argument is exactly a difference-quotient in the limit. Comfort with the integral notation ∫_a^x f(t) dt is all you need; no coding, nothing installed. The practice is pen and paper.

By the end, you’ll be able to

• Define the area function A(x) = integral from a to x of f and explain why it is a function of the moving right end
• Derive A’(x) = f(x) from the single observation that extending the area by dx adds a rectangle of height f(x)
• Explain why A’(x) = f(x) is the fundamental theorem and why integration and differentiation are inverse operations
• Recognize the same pairing in everyday terms (bucket and tap) and in the PDF-and-CDF relationship F’(x) = f(x)

Time and difficulty

• Read time: about 11 minutes
• Practice time: about 12 minutes (verifying A’(x) = f(x), a PDF-and-CDF check, and flashcards)
• Difficulty: standard