Repeated Integrals: Notation & Symbols

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In summary, the conversation discusses the notation for an iterated integral and its use in volume integrals. The symbol "I" is used to represent the integral, and for a hypercubic volume integral in m dimensions, the notation \int_V is used to denote the integration over the region \prod^{m}_{n=0} [0,1].
  • #1
Jheriko
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Is there some notation for an iterated integral, like there is for summations and products?

e.g.

[tex]\overset{m}{\underset{n=0}{\raisebox{-0.07in}{\Huge{\texttt{I}}}}} f(x_0,x_1,x_2,\ldots,x_m) dx_n[/tex]

[tex]\overset{m}{\underset{n=0}{\raisebox{-0.07in}{\Huge{\texttt{I}}}^{b}_{a}}} f(x_0,x_1,x_2,\ldots,x_m) dx_n[/tex]

Here I have used "I" to stand in for whatever the correct symbol might be...

Thanks in advance.
 
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  • #2
If the integral is a volume integral in "n" dimensions (n>=3), then one can put only one integral sign, but specify the fact that the integration is not in one dimension (as one could assume, once he sees only one symbol of integration) through the measure. In this case a volume integral is an iterated integral. Treating hypersurface integrals is not any different, as one could use only one symbol, even if the # of dimensions is not 1.
 
  • #3
So for a 'hypercubic volume integral' over 0 ... 1 in m dimensions, I could do:

[tex]V = \prod^{m}_{n=0} [0,1][/tex]

[tex]\int_V f(x_0,x_1,x_2,\ldots,x_m) dV[/tex]

Thanks.
 

1. What is the meaning of repeated integrals?

Repeated integrals refer to the process of integrating a function multiple times. This can be thought of as the reverse of taking derivatives, where instead of finding the rate of change, we are finding the total accumulation of a function over a specific interval.

2. How is the notation for repeated integrals different from single integrals?

The notation for a single integral is ∫f(x)dx, where f(x) is the function and dx represents the variable of integration. For repeated integrals, the notation will include multiple integrals signs, such as ∫∫f(x,y)dxdy. The number of integral signs will correspond to the number of times the function will be integrated.

3. What do the different symbols in repeated integrals represent?

The integral sign (∫) represents the operation of integration. The function being integrated is represented by the expression inside the integral sign. The variables of integration (dx, dy, etc.) represent the variables with respect to which the integration is being performed.

4. How do the limits of integration work in repeated integrals?

The limits of integration in repeated integrals are written as a range for each variable of integration. For example, if we have the double integral ∫∫f(x,y)dxdy, the limits of integration for x would be written as a range, such as a to b. The limits for y would also be written as a range, such as c to d. This represents the region over which the function is being integrated.

5. What are some common applications of repeated integrals?

Repeated integrals are commonly used in multivariable calculus, physics, and engineering to calculate volume, surface area, and other quantities that require the integration of multiple variables. They are also used in solving differential equations and in statistical analysis.

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