Integration
Derivatives ask "how fast is this changing?" Integration asks the reverse: given how fast something is changing at every moment, how much has accumulated in total? These two questions — developed to solve completely different problems — turn out to be two sides of the same mathematical coin.
Riemann Sums
Start with a concrete problem. A car travels with a velocity $v(t)$ that changes continuously over time. How far does it travel between $t = a$ and $t = b$? If velocity were constant, the answer would be trivial: distance = velocity × time. But it isn't constant — so we approximate.
Divide the time interval $[a, b]$ into $n$ equal pieces of width $\Delta x = \frac{b - a}{n}$. Over each tiny piece, the velocity barely changes — so treat it as constant, equal to the value $f(x_i^*)$ at some chosen sample point $x_i^*$ in that piece. The distance covered in piece $i$ is approximately $f(x_i^*)\,\Delta x$, and the total is approximately:
This is a Riemann sum. Geometrically, each term $f(x_i^*)\,\Delta x$ is the area of a thin rectangle with height $f(x_i^*)$ and width $\Delta x$. The sum approximates the total area between the curve and the $x$-axis on $[a, b]$.
As $n \to \infty$ the rectangles get thinner and the approximation improves. The limit — if it exists — is the definite integral:
The symbol $\int$ is an elongated "S" for "sum." The $dx$ is the infinitesimal width — the limit of $\Delta x$ as $n \to \infty$. Together, $f(x)\,dx$ is the infinitely thin slice of area at position $x$.
A remarkable fact: for any continuous function, it does not matter how you pick the sample points $x_i^*$ — left endpoints, right endpoints, midpoints, random — the Riemann sum always converges to the same value as $n \to \infty$. That value is the integral. The choice of sample points only affects how quickly the sum converges, never where it arrives.
The Integral as Accumulation
Area under a curve is the geometric picture, but the deeper idea is accumulation. The integral adds up infinitely many infinitesimal contributions. The pattern is always: rate × tiny time = tiny contribution; integrate to get the total.
- Velocity (m/s) integrated over time → displacement (m)
- Power (watts) integrated over time → energy (joules)
- Population growth rate integrated over time → total population change
- Rainfall rate (mm/hr) integrated over hours → total rainfall (mm)
When $f(x)$ is negative on part of $[a, b]$, the integral over that region is negative. The definite integral computes net signed area: regions above the $x$-axis contribute positively, regions below negatively. This is not a bug — it is exactly right for computing net displacement, net charge transferred, and so on.
An antiderivative of $f$ is any function $F$ satisfying $F'(x) = f(x)$. The indefinite integral $\int f(x)\,dx = F(x) + C$ represents the entire family of antiderivatives. The constant $C$ is necessary because any constant has derivative zero, so antiderivatives are never unique.
If you drive north for two hours, then south for two hours at the same speed, your displacement is zero — but you've covered real distance. The integral $\int v\,dt$ gives signed displacement; $\int |v|\,dt$ gives total distance traveled. These are different quantities and it matters which one you compute.
The Fundamental Theorem
The Fundamental Theorem of Calculus (FTC) is the central result of the subject. It reveals that differentiation and integration — developed independently to solve different problems — are exact inverses of each other. Newton and Leibniz discovered this in the 1600s, and it unified two centuries of mathematical work overnight.
Part 1 — differentiating an integral:
Define $F(x) = \int_a^x f(t)\,dt$ — the accumulated area from $a$ up to $x$. Part 1 says $F'(x) = f(x)$: if you accumulate $f$ and then differentiate, you recover $f$ exactly. The analogy: $f$ is the speedometer reading and $F$ is the odometer. Differentiate the odometer and you get the speedometer back.
Part 2 — evaluating a definite integral:
Here $F$ is any antiderivative of $f$ — any function with $F' = f$. This is the theorem you use to actually compute integrals. Instead of painfully taking the limit of a Riemann sum, you just find an antiderivative, plug in the two endpoints, and subtract.
Example: $\displaystyle\int_0^2 x^2\,dx$. Since $\frac{d}{dx}\!\left[\frac{x^3}{3}\right] = x^2$, we take $F(x) = \frac{x^3}{3}$ and compute:
Before the FTC, computing the area under a curve required clever geometric arguments invented separately for every single curve. There was no general method. The FTC replaced all of that with a single algorithm: find an antiderivative, evaluate at the endpoints, subtract. It is the reason calculus became the primary tool of physics, engineering, and economics — it made the previously impossible merely routine.