# Moment of Inertia using Triple Integral

• jj2443
In summary, to compute the moment of inertia around the z-axis of a solid unit box with density function \delta=x^{2}+y^{2}+z^{2}, the integral should be set up as \int_{0}^{1}\int_{0}^{1}\int_{0}^{1}(x^{2}+y^{2})(x^{2}+y^{2}+z^{2})dzdydx. The order of integration does not matter in this case.
jj2443

## Homework Statement

Compute the moment of inertia around the z-axis of the solid unit box [0,1]x[0,1]x[0,1] with density given by $\delta$=x$^{2}$+y$^{2}$+z$^{2}$.

## Homework Equations

I=$\int$$\int$$\int$r$^{2}$ $\delta$ dV

## The Attempt at a Solution

I know that the distance r$^{2}$ from the z-axis would be x$^{2}$+y$^{2}$. I don't know how to determine the bounds and the order of the three integrals. Could someone please explain to me how to determine which order I should integrate, and then how I go about finding the bounds of integration for each of the three integrals.

Thanks!

Since none of the boundaries on your volume is dependent on each other you can just write it as a simple 'volume' integral if you want to think of it that way, as if you're finding the volume of the cube but with the other terms involved: $r^{2}$ and ${\delta}$

So in my mind it should be set up as follows:

$$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}(x^{2}+y^{2})(x^{2}+y^{2}+z^{2})dzdydx$$

The order of integration won't matter.

Oh, that's much easier than I was trying to make it. Thank you!

## What is moment of inertia using triple integral?

Moment of inertia using triple integral is a mathematical concept used to calculate the distribution of mass in a three-dimensional object. It involves integrating the mass density function over the volume of the object.

## Why is moment of inertia using triple integral important?

Moment of inertia using triple integral is important because it helps in understanding the rotational behavior of objects. It is used in engineering and physics to analyze the stability, strength, and motion of objects.

## How is moment of inertia using triple integral calculated?

Moment of inertia using triple integral is calculated by integrating the product of the mass density function and the distance from the rotational axis cubed over the volume of the object. This can be expressed as I = ∫∫∫ ρ(x,y,z)(x^2 + y^2 + z^2) dV.

## What are the units of moment of inertia using triple integral?

The units of moment of inertia using triple integral depend on the units of the mass density function and the distances involved. In SI units, it is typically expressed in kg*m^2.

## How does moment of inertia using triple integral differ from other methods of calculating moment of inertia?

Moment of inertia using triple integral is a more general method as it can be applied to objects with irregular shapes and non-uniform mass distributions. Other methods, such as the parallel axis theorem and the perpendicular axis theorem, are specific to certain shapes and require simplifying assumptions.

Replies
1
Views
1K
Replies
2
Views
3K
Replies
1
Views
1K
Replies
6
Views
2K
Replies
5
Views
2K
Replies
1
Views
4K
Replies
3
Views
3K
Replies
5
Views
2K
Replies
5
Views
1K
Replies
2
Views
1K