Integrals and Area

Differentiation breaks things apart — it tells you how a quantity is changing at each instant. Integration puts things back together — it accumulates all those tiny changes into a total. If the derivative answers "how fast?", the integral answers "how much altogether?" This inverse relationship between the two operations is no accident; it is formalized in the Fundamental Theorem of Calculus, one of the most profound results in all of mathematics.

The simplest way to visualize integration is through area. The definite integral $\int_a^b f(x)\,dx$ computes the signed area between the graph of $f$ and the $x$-axis from $x = a$ to $x = b$. Regions above the axis contribute positive area, while regions below contribute negative area. This geometric interpretation makes integrals tangible: you can literally see what the integral measures by shading in the region under a curve.

But area is only the beginning. Integrals compute total distance traveled, total work done by a force, total mass of a varying-density object, and total probability in a distribution. Any time you need to add up infinitely many infinitesimally small contributions, integration is the tool. In physics, the integral of velocity over time gives displacement. In statistics, the area under a probability density curve gives probability. In engineering, the integral of a load distribution gives the total force on a beam.

On the computational side, finding an antiderivative — a function whose derivative is the integrand — is the key skill. Many basic antiderivatives follow directly from reversing the derivative rules you learned in the previous lesson. The power rule reverses into the power rule for integration, the derivative of $\sin x$ reverses to the antiderivative of $\cos x$, and so on. This lesson covers indefinite integrals (antiderivatives), definite integrals (accumulated area), and the Fundamental Theorem that ties them together.