Definite Integrals & the FTC

Riemann Sums - The 30-Second Refresher

$\int_a^b f(x)\,dx$ is the limit of $\sum_{i=1}^n f(x_i^*)\Delta x$ as the partition shrinks. The full pictorial buildup - rectangles narrowing into the area under the curve - lives on the chapter page §4.1 The Definite Integral, which also covers Riemann sums in 2-D and 3-D. We'll skip ahead to the part that's actually a single-variable prerequisite for chapter 4: the Fundamental Theorem.

The Accumulation Function

Here's a question that seems completely different from computing area: if we define a new function $F(x)$ as "the area under $f$ from $a$ up to $x$," what is $F'(x)$?

Think about it for a moment. When we nudge $x$ to $x + h$, the area $F(x)$ grows by a thin strip of width $h$ and height roughly $f(x)$. So $F(x+h) - F(x) \approx f(x) \cdot h$, which means $\frac{F(x+h)-F(x)}{h} \approx f(x)$.

As $h \to 0$, this becomes exact. That's the first version of the Fundamental Theorem of Calculus:

In other words: differentiation and integration are inverse operations. The integral accumulates area; the derivative recovers the original rate.

Below, the top panel shows $f(t) = t^2$ and the shaded area from $0$ to $x$. The bottom panel shows $F(x) = \int_0^x t^2\,dt = \frac{x^3}{3}$, the accumulated area. Drag the slider and notice: the slope of $F$ at each $x$ equals the height of $f$ at that $x$.

The Evaluation Shortcut

The first FTC tells us that $F(x) = \int_a^x f(t)\,dt$ is a primitive of $f$. But if we already know any antiderivative $G$ (where $G' = f$), then $F$ and $G$ differ by a constant. In particular, $F(b) = G(b) - G(a)$, since $F(a) = 0$.

That gives us the second version - the evaluation formula we use every day:

This is the magic that makes calculus practical. Instead of computing a limit of Riemann sums, we just find an antiderivative and subtract. For example:

$$\int_0^{\pi/2} \sin 2x\,dx = \left[-\frac{\cos 2x}{2}\right]_0^{\pi/2} = -\frac{\cos\pi}{2} + \frac{\cos 0}{2} = \frac{1}{2} + \frac{1}{2} = 1$$

Let's put both versions side by side so you can see the full picture:

Properties at a Glance

Linearity, additivity over intervals, the order-flip rule, and the comparison test are all assumed throughout chapters 4 and 5. They live on the chapter page §4.1 Properties of the Definite Integral; if any of them feel unfamiliar, do that card first.