The fundamental theorem of calculus

Differentiation and integration are inverse operations, and the statement of that is the fundamental theorem of calculus¹. We follow it with $f(x) = x^2$ .

Area as a function

Consider the area from $0$ to $x$ as a function of $x$ .

F(x) = \int_0^x t^2 \, dt = \frac{x^3}{3}

Differentiating this $F$ returns $F'(x) = x^2 = f(x)$ . Differentiating the function that represents an area gives back the original function.

\frac{d}{dx} \int_a^x f(t) \, dt = f(x)

The reason appears in how the area grows. When $x$ increases by $h$ , the area gains a thin strip of width $h$ and height about $f(x)$ . The increment is about $f(x) h$ , so dividing by $h$ and letting $h \to 0$ leaves $f(x)$ .

The other form

Writing $F$ for an antiderivative of $f$ gives the method for computing a definite integral.

\int_a^b f(x) \, dx = F(b) - F(a)

Instead of adding up an area in fine pieces, one subtraction of antiderivatives suffices. So $\int_1^2 x^2 dx = \dfrac{8}{3} - \dfrac{1}{3} = \dfrac{7}{3}$ .

Reading the correspondence off the graph

Quantity at $x = 1$	Value
$f(1)$	$1$
$F(1)$	$\dfrac{1}{3}$
Slope of the tangent to $F$ , that is $F'(1)$	$1$

The height of $f$ is the slope of $F$ .

Sign and monotonicity

Sign of $f$	Behavior of $F$
positive	increasing
negative	decreasing
$0$	horizontal tangent

That the sign of $f$ decides whether $F$ rises or falls is another way of stating the theorem.

The parabola on the graph is $y = x^2$ , the cubic representing the area is $y = \dfrac{x^3}{3}$ , and the large dots are the point with $f(1) = 1$ and the point with $F(1) = \dfrac{1}{3}$ .

Fundamental theorem of calculus, Wikipedia