Mathematics

Definite integral - formula in LaTeX

A definite integral is the signed area under the graph between a and b. The limits go in _{a} and ^{b}. The \, command adds a space before the differential, and \mathrm{d} sets the d upright, as the ISO standard recommends.

\int_{a}^{b} f(x)\,\mathrm{d}x

LaTeX code

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$$
\int_{a}^{b} f(x)\,\mathrm{d}x
$$

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What the symbols mean

$a, b$: limits of integration
$f(x)$: integrand
$\mathrm{d}x$: differential of the integration variable

A math formula - one of the most frequently typed in assignments, lecture notes and calculus or algebra homework.

Worked example

The area under the parabola y = x² from 0 to 2. The square bracket with limits is the Newton-Leibniz evaluation; \left[ and \right] size it to the fraction inside.

\int_{0}^{2} x^2\,\mathrm{d}x = \left[\dfrac{x^3}{3}\right]_{0}^{2} = \dfrac{8}{3}

Other ways to write it

An improper integral: infinity in a limit is \infty.
$\int_{0}^{\infty} e^{-x}\,\mathrm{d}x = 1$
```
\int_{0}^{\infty} e^{-x}\,\mathrm{d}x = 1
```
\limits puts the limits above and below the integral sign instead of beside it.
$\int\limits_{a}^{b} f(x)\,\mathrm{d}x$
```
\int\limits_{a}^{b} f(x)\,\mathrm{d}x
```
A double integral over a region D; \iint sets the integral signs with tight spacing.
$\iint_{D} f(x,y)\,\mathrm{d}x\,\mathrm{d}y$
```
\iint_{D} f(x,y)\,\mathrm{d}x\,\mathrm{d}y
```

LaTeX commands in this formula

\int: integral
x_{0}: subscript; wrap several characters in braces
x^{2}: superscript (power); wrap several characters in braces
\,: thin space, e.g. between factors of a product
\mathrm: upright font

How to use this formula in LaTeX?

Copy the code with the button above - $$…$$ is centered display math.
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The formula works without extra packages.

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