Riemann integral

Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable. The Riemann integral of $f$ over $[a, b]$ is the unique number $I$ in the definition of Riemann integrability. It is written $$\int_a^b f(x) \, dx.$$

Remarks

For $f \ge 0$, the integral equals the area under the graph of $f$ and above the $x$-axis. By convention, $\int_a^a f = 0$ and $\int_b^a f = -\int_a^b f$. The fundamental theorem of calculus computes the integral from an antiderivative. The integral is named after Bernhard Riemann.

Explanations

L1 human

The Riemann integral is the limit of sums of areas of thin rectangles under the graph. It gives the signed area between the graph and the $x$-axis.