Change, made visible

Chapter 5 Slope and area undo each other

The fundamental theorem

Differentiation measures slope; integration accumulates area. The fundamental theorem says they are inverse operations. Drag the edge of the area and watch its growth rate equal the height of the curve.

9 min read · 5 interactive figures

Drag x to the right. The green area A(x) grows at a rate equal to the height f(x): the slope of the lower curve is the height of the upper one.

The fundamental theorem of calculus says that differentiation and integration undo each other. Part 1: if A(x) is the area under f from a fixed point a up to x, then the rate at which that area grows is f(x), so A′(x) = f(x). Part 2: to find the area under f from a to b, take any function F whose derivative is f and compute F(b) − F(a).

This site began with two ideas that look unrelated: the slope of a curve at a point, found by shrinking secant lines, and the area under a curve, found by refining rectangles. One is about a single point, the other about a whole interval. One measures change; the other adds it up. The fundamental theorem says they are the same idea run in opposite directions, and it turned calculus from a collection of clever tricks into a machine. This chapter makes the theorem something you can watch happen.

The area so far is a function

Fix a starting point a and let the right-hand edge x move. The signed area under f between them changes as x moves, so it is a function of x, called the accumulation function[1]:

A(x) = ∫ax f(t) dt(5.1)

The letter inside changes from x to t only to avoid a clash: x is now the upper limit, and t runs from a up to it. Think of A as an odometer for area. Where f is positive, moving x to the right adds area and A rises; where f is negative, it subtracts.

Question 1 Predict before you drag.#
Figure 1 Top: the curve f, with the signed area from a to x shaded (hatched where it counts negative). Bottom: that area as a function A(x), with its tangent at x in vermilion. Drag the green handle x. Notice that the tangent’s slope always equals the height of the vermilion bar above: A′(x) = f(x). Drag a and the whole lower curve slides up or down.#

Drag x slowly across the two-tone wave. While the blue curve is high, the green curve climbs steeply; where the blue curve is low but positive, it climbs gently; where the blue curve crosses zero, the green curve levels off at a peak; and where the blue curve is negative, the green curve falls. The height of f sets the steepness of A. The readout confirms it with numbers: the slope of A, measured numerically from the areas, matches f(x) to four decimal places everywhere you put the handle. That is Part 1 of the theorem[1]:

ddx ∫ax f(t) dt = f(x)(5.2)
established

Patrick Thompson’s teaching experiment with senior and graduate mathematics students traced their difficulties with the fundamental theorem to weak images of rate of change and of accumulation, and to never having coordinated the two[2]. The figure puts both on one screen: the bar is a rate, the green curve is an accumulation, and the tangent is where they meet.

Why the slope of the area is the height

The argument fits in one picture. Move x a little to the right, by h. The area grows by a thin strip between x and x + h. That strip is almost a rectangle of width h and height f(x). So

A(x + h) − A(x) ≈ f(x) · h, so A(x + h) − A(x)h ≈ f(x)(5.3)

The left side of the second equation is a secant slope of A, the same quotient as equation (2.2) in the derivative chapter. As h → 0 it becomes the derivative of A, and the approximation becomes exact, because the only error is the curved sliver at the top of the strip, and that sliver shrinks faster than h.

Figure 2 Left: the area up to x (pale green) and the new strip from x to x + h (dark green). Right: the top of the strip, magnified, with the top of the rectangle f(x) · h dashed and the leftover sliver in purple. Shrink h. Notice that the strip’s area divided by h approaches f(x), digit by digit, as the sliver vanishes.#

How fast does the sliver vanish? Its height is about |f′(x)| · h and its width is h, so its area is roughly proportional to h², and dividing by h still leaves something proportional to h. Shrink h by a factor of 10 in the figure and the gap between the two readouts shrinks by about 10 too. A careful proof replaces “almost a rectangle” with an exact statement: for a continuous f there is a point c between x and x + h where the strip’s area is exactly f(c) · h (the mean value theorem for integrals), and as h → 0, c is squeezed onto x[1].

Part 2: areas without rectangles

Part 1 has a consequence that changed mathematics. Suppose you want ∫ab f(x) dx, and you know some function F whose derivative is f, an antiderivative. The area function A is also an antiderivative, by Part 1. Two functions with the same derivative differ by a constant, so A(x) = F(x) + C. At x = a no area has accumulated, so 0 = F(a) + C, and C = −F(a). Then at x = b[1]:

∫ab f(x) dx = F(b) − F(a)(5.4)

The labour of the previous chapter collapses. The area under x² from 0 to 1, which took a sum of squares and a limit in equation (4.2), is now one line: an antiderivative of x² is x³/3, so the area is 1³/3 − 0³/3 = ⅓. The area under one arch of the sine curve, from 0 to π, is −cos π − (−cos 0) = 1 + 1 = 2. An infinite process of finer and finer rectangles is replaced by subtracting two numbers, whenever you can find an antiderivative. Finding antiderivatives, “integration” in the everyday sense, is the reverse of the differentiation rules, and much of a second calculus course is spent on its techniques.

Newton’s picture: a speedometer and an odometer

Newton thought about calculus in terms of motion, quantities flowing in time, and his version of the theorem is the most concrete one. A car’s odometer records its position s(t); the speedometer shows its velocity v(t). Velocity is the rate of change of position, v = s′. And position is accumulated velocity: the distance covered is the area under the velocity graph. These are Parts 1 and 2 of the theorem, in a car.

Figure 3 A car accelerates, brakes, stops at 6 s and then reverses. Press Drive or drag time. Top: the velocity graph, with the area accumulated so far. Bottom: position, with its tangent. Notice that the green area always equals the car’s distance from the start, and that once the car reverses, the area below the axis is subtracted.#

The velocity here is v(t) = ¾t(6 − t) metres per second. An antiderivative is s(t) = ¾(3t² − t³/3), so by equation (5.4) the car is s(6) − s(0) = 27 m from the start when it stops, and s(8) = 16 m from the start two seconds later, after reversing 11 m. The integral gives displacement, the net change. The odometer in a real car counts total distance instead, 27 + 11 = 38 m, which is the integral of the speed |v|, not of the velocity v. Students who learn that “the integral is the area under the curve” often cannot say which area a physical question needs[3]; this car is the standard example of why it matters.

documented

Each undoes the other, almost

“Inverse operations” needs one qualification. Integrate first and then differentiate, and you get back exactly what you started with: that is equation (5.2). Differentiate first and then integrate, and you get back what you started with up to a constant, because differentiation throws constants away. The functions x², x² + 5 and x² − π all have derivative 2x, and nothing in 2x remembers which one you began with.

Figure 4 Three steps. Start with f (blue). Integrate it from a to get A (green; pale green curves are other starting points). Differentiate A numerically (vermilion), and f reappears underneath, exactly. Drag a: A moves up and down by a constant, and step 3 does not change at all.#

This is why indefinite integrals come with “+ C.” The family of all antiderivatives of f is a stack of identical curves shifted vertically, the pale curves in the figure, and choosing the starting point a of the accumulation picks one of them. For a definite integral the constant cancels in F(b) − F(a), which is why any antiderivative will do in equation (5.4).

Two ideas, one subject

Look back at the five chapters. Limits made it possible to talk about a slope at a single point and an area made of infinitely thin rectangles. The derivative measures how fast a quantity is changing; the chain rule says that when quantities drive one another, their rates multiply. The integral adds up a rate to recover the total change. And the fundamental theorem says that adding up rates and measuring rates are inverse processes: the area under a speed graph is a distance, and the slope of a distance graph is a speed.

Almost every use of calculus in science goes through this theorem. A differential equation states a rate of change, as Newton’s laws do for motion, and solving it means accumulating that rate to find the quantity itself. When the rate has no tidy antiderivative, as with the bell curve in the previous chapter, the accumulation is done numerically, with the same kind of adaptive rectangles you have been dragging. Either way the idea is the one on the home page: slope and area, each undoing the other.