# Triple integral depending on a parameter

• Draconian28
In summary, the integral \iiint (x^{2n} + y^{2n} + z^{2n})\,dV can be evaluated by using spherical polar coordinates and taking advantage of symmetry and the formula for integrals of powers of sine and cosine. For n = 1, the integral represents the sum of inertias of a sphere with unit density about the three coordinate axes.
Draconian28

## Homework Statement

Find $$\iiint (x^{2n} + y^{2n} + z^{2n})\,dV$$ where the integral is taken over the region of 3D space where $$x^{2} + y^{2} + z^{2} \leq 1$$

## The Attempt at a Solution

I tried doing this in Cartesian coordinates, but the limits of integration got very messy and I got stuck after doing the first integral. I also tried using spherical polar coordinates, and then the limits of integration are quite simple, but the integrand gets complicated, unless $n = 1$, in which case the integral is quite easy to do.

I then thought that, since the only case where this looks simple enough to do directly is $n = 1$, I could try to make a conjecture as to what the value of the integral is for general n and then try to prove it by induction. The problem with that, though, is that I don't see how to go from the case $n = k + 1$ to the case $n = k$.

Any ideas?

I haven't tried it myself yet, and the wife wants the computer just now, but have you tried spherical coordinates? That would be the natural first choice. May or may not work...

[Edit later] I think you can work it with spherical coordinates using symmetry and the formula$$\int_0^\frac \pi 2 \sin^n x\, dx =\int_0^\frac \pi 2 \cos^n x\, dx = \frac{1\cdot 3\cdot 5\cdot\cdot\cdot(n-1)}{2\cdot 4\cdot 6\cdot\cdot\cdot n}\frac \pi 2$$which is valid for ##n## even and ##n \ge 2##.

Last edited:
This integral covers the volume enclosed by a sphere with radius = 1. It can be broken up into the sum of its parts, since the integrand is a sum.

For n = 1, the integral will evaluate to the sum of the inertias of a sphere about the three coordinate axes, assuming a unit density for the sphere.

## 1. What is a triple integral depending on a parameter?

A triple integral depending on a parameter is a mathematical concept that involves integrating a function of three variables over a three-dimensional region, where one of the variables is a parameter that can take on different values. This type of integral is commonly used in physics and engineering to solve problems involving changing parameters, such as time or position.

## 2. How is a triple integral depending on a parameter different from a regular triple integral?

In a regular triple integral, all three variables are independent and the integration is performed over a fixed region. However, in a triple integral depending on a parameter, one of the variables is dependent on a parameter and the integration is performed over a variable region that changes with the parameter values.

## 3. What is the purpose of using a triple integral depending on a parameter?

The purpose of using a triple integral depending on a parameter is to solve problems that involve changing parameters. This allows for a more flexible and accurate solution, as the integration can take into account the changing values of the parameter.

## 4. What types of problems can be solved using a triple integral depending on a parameter?

A triple integral depending on a parameter can be used to solve a wide range of problems, such as finding the volume of a changing shape, calculating the work done by a changing force, or determining the mass of a changing object. It is particularly useful in physics and engineering applications.

## 5. What are some examples of triple integrals depending on a parameter?

One example of a triple integral depending on a parameter is calculating the work done by a force that changes with time. Another example is finding the volume of a solid that changes shape as a parameter, such as temperature, varies. Other examples include determining the center of mass of a changing object and calculating the moment of inertia of a rotating body with changing dimensions.

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