Is there a simple rule for double and multiple integration?

In summary, the rules of integration are based on single integrals and can be extended to double and multiple integrals through the use of iterated integrals. There are no special rules for double integrals, etc. and they can be evaluated through a series of single integrals. Antiderivatives and integrals are not the same thing, and to evaluate a double integral analytically, iterated integrals are typically used.
  • #1
JizzaDaMan
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Hi, I have a very limited knowledge of calculus, and even less of integration. However I know that the general rule for integration is
x[itex]^{n}[/itex] integrates as [itex]\frac{x^{n+1}}{n+1}[/itex]

Is there a similar rule for double integration and is there a rule that can be extended to multiple integrals beyond double?
 
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  • #2
The rules of integration are developed assuming a single integral. Double integrals and other iterated integrals can be evaluated by means of a series of single integrals.

In other words, there are no special rules of integration for double integrals,etc.
 

1. What is a double integral?

A double integral is a type of integration that involves calculating the area under a surface in a two-dimensional space. It is used to find the volume of three-dimensional objects or to solve problems involving two variables.

2. How do I set up a double integral?

To set up a double integral, you need to determine the limits of integration for both variables. This can be done by visualizing the region of integration and identifying the boundaries. The integrand, which is the function being integrated, is then multiplied by the infinitesimal area element (dx dy or dA) and integrated over the limits of both variables.

3. What is the difference between a single and a double integral?

A single integral involves calculating the area under a curve in a one-dimensional space, while a double integral involves calculating the area under a surface in a two-dimensional space. In a single integral, there is only one variable to integrate over, while in a double integral, there are two variables to integrate over.

4. What is the purpose of using a double integral?

A double integral is used to find the volume of three-dimensional objects or to solve problems involving two variables. It can also be used to calculate areas and moments of inertia.

5. How do I solve a double integral?

To solve a double integral, you can use different integration techniques such as substitution, integration by parts, or partial fractions. It is important to carefully set up the limits of integration and to integrate with respect to the correct variables.

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