Which Integration Technique Should I Use for This Triple Integral?

In summary, the conversation is about the best method to use for integrating the given expression, which involves a triple integral with limits for R, theta, and phi. The expert suggests doing the theta integration first, taking note of the sine outside the root, and being careful with absolute values in the R integration. They also advise against using integration by parts and suggest using a helpful expression for the integrand.
  • #1
MathematicalPhysics
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[tex]\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 \sin\theta\,dR\,d\theta\,d\phi[/tex]

with the integration over R between 0 and a
the integration over between 0 and pi
the integration over between 0 and 2pi

Should I use integration by parts or should I take the R^2 sin(theta) under the square root?

Any hints and tips are much appreciated!
 
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  • #2
Do the [tex]\theta[/tex] integration first (note that the sine outside the root makes this easy).

Beware absolute values in your R integration!
 
  • #3
Do I need to change the order of integration then and have new limits or can I choose to rearrange it to a more convenient form, like

[tex]\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 sin\theta\,d\theta\,dR,d\phi[/tex]

then integrating wrt [tex]\theta[/tex] by parts?
 
  • #4
You don't do it by parts!

Note that your integrand equals:
[tex]\frac{\partial}{\partial\theta}\frac{R}{3a}(R^{2}-2aR\cos\theta+a^{2})^{\frac{3}{2}}[/tex]
 

What is a triple integral?

A triple integral is a mathematical calculation that involves finding the volume of a three-dimensional shape or region. It is a type of multivariable integral in which the integration is performed over a three-dimensional space.

Why is it called "triple" integral?

The term "triple" refers to the fact that the integral involves three variables, typically denoted by x, y, and z. This is in contrast to a single integral, which involves one variable, or a double integral, which involves two variables.

What is "triple integral trouble"?

"Triple integral trouble" is a colloquial term used to describe the difficulties that can arise when solving a triple integral. This can include complex mathematical calculations, challenging integration techniques, and the potential for making mistakes.

When is a triple integral used?

Triple integrals are commonly used in mathematics, physics, and engineering to solve problems involving three-dimensional shapes or volumes. They are also used in probability and statistics to calculate probabilities in three dimensions.

What are some strategies for solving a triple integral?

There are several techniques that can be used to solve a triple integral, including changing the order of integration, using symmetry, and using specific integration rules such as the substitution rule. It is also important to carefully set up the integral and choose appropriate limits of integration.

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