Definite Integrals and Areas

Integration was born from a geometric problem: finding the area of a region with a curved boundary. The definite integral solves this, and the astonishing link between area and antidifferentiation is the crowning result of introductory calculus.

The definite integral. The definite integral of $f$ from $a$ to $b$ is written $\int_a^b f(x)\,dx$, where $a$ and $b$ are the limits of integration. Conceptually it is the limit of a sum of thin rectangles of width $\Delta x$ and height $f(x)$ that fill the region under the curve — this is why the symbol is an elongated S, for "sum." Unlike an indefinite integral, a definite integral evaluates to a single number, and no $+C$ appears.

The Fundamental Theorem of Calculus. The theorem that ties everything together states that if $F$ is any antiderivat