How to Write Triple Integral (∫∫∫) Symbol in LaTeX

To write triple integral symbol (∫∫∫) in LaTeX, use the LaTeX command \iiint. It will add ∫∫∫ symbol in the text.

In this article, we will discuss how to use calculus double integral symbol (∫∫∫) in the LaTeX document and its use to represent the integration of a three-dimensional function over a region in space.

Symbol Overview

Symbol: Triple Integral

Unicode: U+222D

Type: Mathematical

Package Requirement: None (built-in symbol)

Argument: None (no additional arguments needed)

LaTeX Command: \iiint

Example: ∫∫∫

Description

The triple integral symbol (∫∫∫) is built-in symbol available in LaTeX,denoted by command \iint. It is used to represent triple integral in mathematics. It is widely used in calculus, physics, and engineerign that involves three-dimensional systems.

Syntax

The LaTeX command \iiint is used to write the triple integral symbol.

\iiint

Using the Triple Integral Symbol in Cartesian Coordinates

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[ \iiint_R (x^2 + y^2 + z^2) \, dV \]
\end{document}

Output 👇️

∫∫∫_R(x^2 + y^2 + z^2)dV

In this example, the triple integral symbol is calculated over the region R, and the integrand is a function of the coordinates x,y, and z.

Using the Triple Integral Symbol in Cylindrical Coordinates

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[ \iiint_S (r\cos\theta)^2 \, r \, dr \, d\theta \, dz \]
\end{document}

Output 👇️

∫∫∫_S(rcosθ)2rdrdθdz

Here, in the above example, the triple integral symbol is computed over a cylindrical region S and the integrand is represented in terms of cylindrical coordinates (r,θ,z).

Using the amsmath Package to Use triple Integral Symbol

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[ \iiint_T \rho^2 \sin\phi \, d\rho \, d\theta \, d\phi \]
\end{document}

Output 👇️

∭_Tρ^2sinϕdρdθdϕ

In the above example, it uses the amsmath package for mathematical typesetting. In LaTeX document, type \usepackage{amsmath} to use triple integral symbol in your document. The triple integral symbol is computed over a spherical region T, and the integrand is given in terms of spherical coordinates (ρ,θ,ϕ).

Applications

The triple integral symbol has extensive applications various mathematical and scientific fields such as:

• Calculate volume of three-dimensional objects and regions.
• Calculation of mass, and moment of inertia of solid bodies.

Conclusion

The triple integral symbol (∫∫∫) is a fundamental mathematical symbol used widely in calculus,physics and engineering.

Using the above syntax and examples presented in this article, you can write and use triple integral symbol in LaTeX documents to represent the integration and solve complex mathematical expressions.