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# Lesson 11.9: Integrals

The elongated letter s is the most commonly used notation for integral or indefinite integral. The braille symbol for the integral sign is not a letter; therefore, it is not treated as a letter, but as a mathematical symbol.

## Brailling the integral

There are no spaces between the braille symbol for the integral, dots two three four six, and the expression associated with it, including the final portion of the expression, often written as dx or dt. Some material transcribed prior to an errata published by BANA may show the spaces.

$\int$

## There can also be double and triple integrals

Material preceding the integral sign in an expression is not included in the five-step process for modification.

## Material directly under and over the integral sign

The integral sign may appear with material directly over it and under it or may have material that is superscripted and subscripted. In addition, there may be a bracket after the expression that contains superscripted or subscripted material.

The material that is displayed directly under is brailled after the integral symbol and is followed by the material displayed directly over. This is similar to what is done when brailling sigma notation.

## Material subscripted and superscripted to the integral sign

The general guidelines for superscripted or subscripted expressions apply to superscripts or subscripts to the integral sign. Numeric subscripts require the subscript indicator because the integral sign is not a letter. The subscript is brailled first and is also the starting value of the interval. The superscript is brailled second and is also the ending value of the interval. There is no baseline indicator between the subscripts and superscripts of the interval. There is a dot five following the superscript. When the integral is presented this way, you should not use a termination sign.

Multiple integrals may each have their own subscripts and superscripts.

## Integrals modified by superimposed symbols

The integral sign may have a variety of shapes or symbols superimposed on it. In print, a circle, square, rectangle, or a sign for infinity may appear in the center of the integral sign. In braille, the modification is displayed with the integral sign first, dot four to indicate there is a superimposed symbol, the symbol or symbols that are superimposed, and the termination indicator to signal the end of the modification. The modified integral sign is punctuated and spaced in the same manner as an unmodified integral sign.

### Example 1

$\int {x}^{2}d\mathbf{x}$
⠮⠭⠘⠆⠐⠙⠭

### Example 2

$\int {x}^{2}d\mathbf{x}=\frac{1}{3}{x}^{3}+C$
⠮⠭⠘⠆⠐⠙⠭⠀⠨⠅⠀⠹⠂⠌⠒⠼⠭⠘⠒⠐⠬⠠⠉

### Example 3

$\int \left({x}^{2}-2x+4\right)d\mathbf{x}$
⠮⠷⠭⠘⠆⠐⠤⠆⠭⠬⠲⠾⠙⠭

### Example 4

$\int \int f\left(x, y\right)\mathrm{dA}$
⠮⠮⠋⠷⠭⠠⠀⠽⠾⠙⠠⠁

### Example 5

$\underset{1}{\overset{\infty }{\int }}3{x}^{2}\mathrm{dx}$
⠐⠮⠩⠂⠣⠠⠿⠻⠒⠭⠘⠆⠐⠙⠭

### Example 6

$5\underset{a}{\overset{b}{\int }}f\left(x\right)\mathrm{dx}$
⠼⠢⠐⠮⠩⠁⠣⠃⠻⠋⠷⠭⠾⠙⠭

### Example 7

${\int }_{0}^{2}f\left(x\right)\mathrm{dx}$
⠮⠰⠴⠘⠆⠐⠋⠷⠭⠾⠙⠭

### Example 8

${\int }_{0}^{\infty }\mathrm{sin}\mathrm{xdx}$
⠮⠰⠴⠘⠠⠿⠐⠎⠊⠝⠀⠭⠙⠭

### Example 9

${\int }_{0}^{1}{\int }_{0}^{2x}\left(x+2y\right)\mathrm{dydx}$
⠮⠰⠴⠘⠂⠐⠮⠰⠴⠘⠆⠭⠐⠷⠭⠬⠆⠽⠾⠙⠽⠙⠭

### Example 10

${\oint }_{C}\mathrm{Mdx}+\mathrm{Ndy}$
⠮⠈⠫⠉⠻⠰⠠⠉⠐⠠⠍⠙⠭⠬⠠⠝⠙⠽