Integrals accumulate · Math Foundry

The idea

An integral adds infinitely thin slices together to measure total accumulation. Rectangles under a curve give a rough estimate of the area; using more, thinner rectangles improves that estimate, and in the limit it becomes exact. The fundamental theorem of calculus links this accumulation back to derivatives — integration and differentiation are inverse operations.

Real-world: tracking a model rocket

The height sensor fails, but velocity is recorded as $v(t) = 3t^2$ metres per second; integrating velocity from $0$ to $2$ seconds reconstructs the rocket's displacement. $\int_0^2 3t^2 \, dt = \left[t^3\right]_0^2 = 8$ — that 8 is the exact area under the velocity curve, so the rocket travels 8 metres in those two seconds.

6 rectangles. More slices → less empty space → a better estimate.

Worked example

For $f(x)=x$ on $[0,2]$ , the region under the curve is a triangle:

\begin{aligned} \text{Area} &= \frac{1}{2}\times 2\times 2 = 2, \\ \int_0^2 x\,dx &= 2. \end{aligned}

Cheat sheet

$\int_a^b f(x)\,dx = \text{accumulated signed area}$

More, thinner rectangles in a Riemann sum converge toward the exact integral.
The Fundamental Theorem of Calculus: integration undoes differentiation, and vice versa.
Integrating a rate (like velocity) over time gives total accumulated change (like displacement).
Area below the $x$ -axis counts as negative — integrals track signed area, not just magnitude.