Here is a strange fact that is easy to miss once you've been drilled on it: nothing about slicing a region into rectangles and adding up their areas should have anything to do with finding the slope of a tangent line. One process accumulates; the other localizes. One asks "how much, in total, over this whole interval?" The other asks "how fast, right at this instant?" And yet these two constructions—built independently, for different reasons, using entirely different limiting arguments—turn out to undo each other exactly. That fact has a name: the Fundamental Theorem of Calculus. It is not a computational trick. It is a genuine, surprising theorem, and Michael Spivak's Calculus devotes its Chapter 14 to proving it with the care it deserves.
Two Constructions That Shouldn't Be Related
By the time you reach this chapter, you've already built two separate towers. The derivative comes from Chapter 10: it's a limit of difference quotients, capturing instantaneous rate of change. The integral comes from Chapter 13: it's built from lower and upper sums—inf and sup of a function over tiny subintervals—capturing accumulated area, defined so carefully that Spivak spends an entire chapter just proving continuous functions are integrable at all.
Nothing in the definition of the integral mentions derivatives. Nothing in the definition of the derivative mentions area. You could imagine a mathematical universe where the "area function" of a curve is some wild, unrelated object with no clean relationship to the curve's slope. The Fundamental Theorem of Calculus is the statement that we don't live in that universe.
The Theorem Itself
Spivak states it like this. Let f be integrable on [a, b], and define a new function F by letting its value at x be the accumulated area from a up to x:
F(x) = ∫ax f
Theorem 1 (The First Fundamental Theorem of Calculus). If f is continuous at c in [a, b], then F is differentiable at c, and
F′(c) = f(c).
Read that again slowly, because it's the whole surprise packed into four symbols. F was built by accumulating area—a global, cumulative process that in principle depends on the entire history of f from a to c. Yet its derivative at c—a purely local, instantaneous quantity—turns out to equal nothing more exotic than the height of the original curve at that single point. All that accumulated information collapses, under differentiation, back down to one number: f(c) itself.
Why It's True: The Squeeze at the Heart of the Proof
The proof is a beautiful example of squeezing a quantity between two bounds that both converge to the same limit. Consider F(c + h) − F(c), which is just the area of the thin sliver from c to c + h. Let mh be the smallest value f takes on that sliver, and Mh the largest. Since the sliver's area is trapped between the areas of the inscribed and circumscribed rectangles built from those extremes,
mh · h ≤ F(c + h) − F(c) ≤ Mh · h,
which after dividing by h gives
mh ≤ [F(c + h) − F(c)] / h ≤ Mh.
Now comes the one place continuity earns its keep: because f is continuous at c, as h → 0 both mh and Mh are forced to converge to f(c)—the sliver gets so thin that its highest and lowest values both squeeze down onto the value right at c. The difference quotient defining F′(c) is trapped between two things converging to f(c), so it converges there too. That's the entire argument (Spivak handles h < 0 with a mirrored version). No exotic machinery—just a squeeze, and the humble observation that a continuous function can't jump around on a shrinking interval.
A Worked Example, and Where the Payoff Lands
The First Fundamental Theorem tells you F is a derivative of f. Turn that around and you get the tool everyone actually uses to compute integrals—Spivak's Corollary (elevated to the Second Fundamental Theorem once it's proved to hold even for merely integrable, not-necessarily-continuous, f): if f = g′ for some function g, then
∫ab f = g(b) − g(a).
Here's the concrete case. Take f(x) = x². You happen to know a function whose derivative is x², namely g(x) = x³/3, since g′(x) = x² by the power rule. The theorem says you can skip lower and upper sums entirely:
∫ab x² dx = b³/3 − a³/3.
The same reasoning gives ∫ab xn dx = (bn+1 − an+1)/(n + 1) for any natural number n—a formula that would otherwise demand the same laborious sum-of-powers estimates used to define the integral in the first place. What used to take pages of inequalities now takes one line, because differentiation and integration are inverse operations.
But notice the theorem cuts both ways, and this is where it gets genuinely useful rather than just tidy: it also manufactures functions you didn't have before. Spivak points out that if f(x) = 1/x for x > 0, then
g(x) = ∫1x (1/t) dt
satisfies g′(x) = f(x)—and "we know of no simpler function g with this property," as the text puts it. The integral isn't just a computational shortcut here; it's the only tool that hands you this function at all (it will eventually earn the name ln x, but you don't need to know that yet to see the point). Every continuous function, no matter how unfamiliar, is secretly the derivative of something, and the integral is what constructs that something.
Building on the Theorem: Chain Rule Meets Accumulation
Once F is known to be differentiable with F′ = f, it behaves like any other differentiable function under composition. Spivak works through functions like f(x) = ∫ax³ 1/(1 + sin² t) dt, which is really F composed with the cubing function—so the Chain Rule hands you f′(x) = 3x²/(1 + sin² x³) without ever needing to know what the antiderivative of 1/(1 + sin² t) actually looks like. That's the real power of Theorem 1: it lets you differentiate through an integral sign as casually as through any other function, because the theorem has already promised you that the integral is differentiable, and told you exactly what its derivative is.
If this kind of argument—careful, surprising, and ultimately clarifying rather than just formula-generating—is what you want more of, Chapter 14 is worth reading in full, proofs included, alongside the rest of Spivak's Calculus, 4th Edition, which builds the entire theory of derivatives and integrals from the ground up before it ever lets you take a shortcut.
Further reading: How to self-study Spivak's Calculus · Table of contents · Index · Calculus, 4th Edition · Answer Book
Before you read the proof above, could you have guessed why continuity of f at a single point is exactly the ingredient needed to make F′(c) = f(c) true there?
