The Fundamental Theorem of Calculus

Thirty-two midpoint rectangles under x² from 0 to 2 still miss by 0.0007. The antiderivative x³/3 gives the exact 8/3 in one subtraction: 8/3 − 0.

Two ideas, one theorem

So far, derivatives and integrals have looked like separate subjects: one about slopes, the other about areas. The Fundamental Theorem of Calculus says they are inverse processes. OpenStax describes the relationship as discovered and explored by both Isaac Newton and Gottfried Wilhelm Leibniz, among others, during the late 1600s and early 1700s. The theorem has two parts, and the second part is the one that makes integrals easy to compute.

Antiderivatives

An antiderivative of f is a function F whose derivative is f. Since d/dx x³/3 = x², the function x³/3 is an antiderivative of x². So is x³/3 + 7, or x³/3 minus any constant, because constants differentiate to zero. The general antiderivative is written ∫ f(x) dx = F(x) + C, an indefinite integral, with C standing for any constant. Reading the derivative rules backwards gives a short table: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, ∫ 1/x dx = ln|x| + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ e^x dx = e^x + C.

Part 2: evaluating integrals

If f is continuous on [a, b] and F is any antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). The difference is often written [F(x)]_a^b. For ∫₀² x² dx, take F(x) = x³/3: F(2) − F(0) = 8/3 − 0 = 8/3, the value the rectangles were creeping toward. The constant C does not matter because it cancels in the subtraction. Other examples: ∫₁³ 2x dx = 9 − 1 = 8; ∫₀^π sin x dx = (−cos π) − (−cos 0) = 1 + 1 = 2; ∫₁⁴ √x dx = (2/3)(8 − 1) = 14/3.

Compare the effort. The lab below needs 64 rectangles to get the midpoint sum within a few ten-thousandths of 8/3, and it still never lands on it exactly. The theorem gives the exact value with two evaluations of x³/3. That trade, finding one antiderivative instead of adding endless rectangles, is why the theorem carries the name fundamental.

Part 1: the area function

Part 1 explains why antiderivatives appear. Define an area function A(x) = ∫_a^x f(t) dt, the area from a fixed start a up to a moving end x. The theorem says A′(x) = f(x). Picture it: when x moves a little, by h, the area grows by a thin strip of width h and height about f(x), so (A(x + h) − A(x))/h is about f(x), exactly in the limit. The rate at which the area grows is the height of the curve at the moving edge. So the area function is an antiderivative, and Part 2 follows from it.