Triple Integral

The Mathematical Symbol "Triple Integral (∭)"

The ∭ Symbol: Triple Integral

The ∭ symbol signifies a triple integral, which is used in mathematics to compute the integral of a function over a three-dimensional domain. This domain is typically a subset of ℝ³ (the set of all ordered triples of real numbers). Triple integrals find significant applications in physics, especially when calculating physical quantities over a volume, such as mass, electric charge, and momentum. In this article, we will briefly discuss its usage and provide a couple of illustrative examples.

Usage

The triple integral is often employed when working with functions of three variables. It is used to integrate a function over a three-dimensional region. The region of integration can be defined using either Cartesian, cylindrical, or spherical coordinates.

Examples

• Example 1: Volume Calculation
  To determine the volume of a solid region \( E \) under the surface \( z = f(x,y) \) and above the xy-plane, the triple integral can be represented as:
  \( \text{Volume} = \iiint_E dV \)
  where \( dV \) is a differential volume element in the region \( E \).
• Example 2: Mass Calculation for Variable Density
  If a solid region \( E \) has a density function \( ρ(x,y,z) \), the total mass of the region is given by:
  \( \text{Mass} = \iiint_E ρ(x,y,z) \, dV \)
  Here, the triple integral calculates the sum of the masses of all the infinitesimally small volume elements \( dV \) in \( E \).

In summary, the ∭ symbol, denoting the triple integral, is a crucial mathematical tool for various applications in both pure mathematics and physics, especially when dealing with three-dimensional domains. Its understanding is essential for advanced studies in multivariable calculus and its applications.

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Codes for the ∭ Symbol

The Symbol	∭	copy
Alt Code	Alt 8749	copy
HTML Code	∭	copy
HTML Entity	∭	copy
CSS Code	\222D	copy
Hex Code	∭	copy
Unicode	U+222D	copy

How To Insert the ∭ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ∭ symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8749. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML Text	Output
<b>My symbol: &#8749;</b>	My symbol: ∭

(Method 4) Use the HTML Entity Code (for webpages).

HTML Text	Output
<b>My symbol: &iiint;</b>	My symbol: ∭

(Method 5) Use the CSS Code (for webpages).

CSS and HTML Text	Output
<style> span:after { content: "\222D";} </style> <span>My symbol:</span>	My symbol: ∭

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML Text	Output
<b>My symbol: &#x222D;</b>	My symbol: ∭

On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x222D to place the ∭ symbol on your canvas. For example:

JavaScript Text
const x = "0x"+"E9" ctx.fillText(String.fromCodePoint(x), 5, 5);
Output
∭

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ∭ is U+222D. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:

Type	Output
222D [Hold down Alt] [Press x]	∭ (The 222D turns into ∭. Note that you can omit any leading zeros.)

In JavaScript, the syntax is \uXXXX. So, our example would be \u222D. (Note that the format is 4 hexadecimal characters.)

JavaScript Text	Output
let str = "\u222D" document.write("My symbol: " + str)	My symbol: ∭

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