Students often meet the Fundamental Theorem of Calculus as an evaluation rule, \(\textstyle\int_a^b f(x)\,dx = F(b)-F(a)\), where \(F'=f\). That rule is important, but it is not the most revealing way to understand the theorem, and the deeper statement is that accumulation and differentiation are inverse operations.

Build a function out of area

Let \(f\) be continuous and define

\[A(x) = \int_0^x f(t)\,dt\]

The variable \(t\) belongs inside the integral because \(x\) is already serving as the upper limit. The function \(A\) takes an input \(x\) and returns the signed accumulation from 0 to \(x\). If \(f\) is positive, \(A\) increases, and if \(f\) is negative, \(A\) decreases. And \(A(0)=0\).

Watch the accumulation

Top: the rate function f, with the region from 0 to x shaded: dark where it counts positively, light where it counts negatively. Bottom: the accumulation function A(x), traced as far as the sweep has gone, with its tangent line drawn at the leading point. Watch three things. A rises while f is above the axis and falls while f is below. A peaks at the exact moment f crosses zero. And the slope of the tangent always matches the height of f in the top panel. That last observation is the Fundamental Theorem.

The upper graph shows \(f\) and the lower graph shows \(A\). As \(x\) moves to the right, the shaded region in the upper panel records the signed area accumulated so far, and at the same time the lower graph traces the value of \(A(x)\).

Three things are worth watching. When \(f\) is above the axis, \(A\) rises, when \(f\) is below the axis, \(A\) falls, and the slope of \(A\) at each point matches the height of \(f\) there. That last observation is the first part of the Fundamental Theorem.

Why \(A'(x)=f(x)\)

Consider a small increase from \(x\) to \(x+h\). The change in accumulated area is \(A(x+h)-A(x)\), and for small \(h\) this additional region is close to a rectangle with width \(h\) and height \(f(x)\), so \(A(x+h)-A(x) \approx f(x)h\).

Divide by \(h\) to get \(\tfrac{A(x+h)-A(x)}{h} \approx f(x)\). As \(h\to0\), continuity makes the approximation exact, therefore

\[A'(x) = f(x)\]

The rate at which signed area accumulates is the value of the function being accumulated, and that is the central connection between integration and differentiation.

The evaluation rule follows

Now suppose \(F\) is any antiderivative of \(f\). Then \(A\) and \(F\) have the same derivative, \(A'=F'=f\), so they differ by a constant. Write \(F(x)=A(x)+C\), and then \(F(b)-F(a) = A(b)-A(a)\). But \(A(b)-A(a) = \textstyle\int_a^b f(t)\,dt\), therefore

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

The familiar evaluation rule follows from the accumulation-function statement, and the constant disappears because subtraction removes it.

Three common forms

An accumulation function

Suppose \(g(x) = \textstyle\int_2^x f(t)\,dt\). Then \(g'(x)=f(x)\), and, where \(f\) is differentiable, \(g''(x)=f'(x)\). Values of \(g\) come from signed area, and derivatives of \(g\) come from the graph of \(f\).

A variable upper limit

If the upper limit is itself a function of \(x\), use the chain rule. For example, \(\textstyle\tfrac{d}{dx}\int_0^{x^2}\cos(t)\,dt = \cos(x^2)\cdot2x\). The Fundamental Theorem evaluates the integrand at the upper limit, and the chain rule contributes the derivative of that upper limit.

Net change

If \(f'\) is a rate of change, then \(f(b) = f(a) + \textstyle\int_a^b f'(t)\,dt\). This is the net change form, and a starting amount plus accumulated rate gives the final amount. If a tank begins with 30 gallons and water enters at rate \(R(t)\), then after 6 units of time the amount is \(30+\textstyle\int_0^6R(t)\,dt\). No antiderivative formula is required if the integral is evaluated numerically.

A useful self-test

Sketch any continuous function \(f\), then sketch \(A(x) = \textstyle\int_0^x f(t)\,dt\) without finding an antiderivative, using only the graph of \(f\). Where \(f\) is positive, \(A\) should rise, and where \(f\) is negative, \(A\) should fall. Where \(f=0\), \(A\) may have an extremum, and where \(f\) is increasing, \(A\) should be concave up. If you can build that second graph from the first, the theorem is doing more than supplying an integration shortcut.