# Triple Integral For Moment Of Inertia

• Air
In summary, the conversation discusses a solid bounded by the coordinate plane and a plane with a given equation. The question clarifies the limits for each axis and asks about the purpose of the given value in the equation. The responder provides a visual representation and asks if the person has taken multivariable calculus. The conversation then continues to discuss the density function and clarification on the axis mentioned in the question.
Air
I have general question which need to be answered before I can understand steps which I have to do. There are:

• When you are told that a solid is bounded by the coordinate plane and the plane $$x+10y + 2z = 5$$, are the limits considered to be $$0-1$$ for $$x$$-axis, $$0-10$$ for the $$y$$-axis and $$0-2$$ for the $$z$$ axis. What is the $$=5$$ used for in this question?
• If you are told that the density is directly proportional to the distance from the $$y-z$$ axis, does that mean that the density is $$kx$$?
Could you help clear my mind? Thanks in advance.

Last edited:
Air said:
• When you are told that a solid is bounded by the coordinate plane and the plane $$x+10y + 2z = 5$$, are the limits considered to be $$0-1$$ for $$x$$-axis, $$0-10$$ for the $$y$$-axis and $$0-2$$ for the $$z$$ axis. What is the $$=5$$ used for in this question?
Draw a picture. The solid in question is tetrahedral in shape, while the limits you give describe a rectangular brick. The equation they gave describes a plane with normal vector (1, 10, 2) that intersects the axes at the points (5, 0, 0), (0, 1/2, 0) and (0, 0, 5/2). Have you taken multivariable calculus yet?

• If you are told that the density is directly proportional to the distance from the $$y-z$$ axis, does that mean that the density is $$kx$$?

Did they say y-z axis, y=z axis or yz-plane ? I'm not sure what they could mean by y-z axis.

slider142 said:
Draw a picture. The solid in question is tetrahedral in shape, while the limits you give describe a rectangular brick. The equation they gave describe a plane with normal vector (1, 10, 2) that intersects the axes at the points (5, 0, 0), (0, 1/2, 0) and (0, 0, 5/2). Have you taken multivariable calculus yet?

Yes, I've taken multivariable calculus. I understand the further process to work out the moment of inertia but the limits and the density function that I have to insert confuses me.

slider142 said:
Did they say y-z axis, y=z axis or yz-plane ? I'm not sure what they could mean by y-z axis.

Sorry for the confusion. It said $$y-z$$ plane.

## What is a triple integral for moment of inertia?

A triple integral is a mathematical concept used in calculus to find the moment of inertia of a three-dimensional object. Moment of inertia is a measure of an object's resistance to rotational motion.

## How is a triple integral for moment of inertia calculated?

The triple integral for moment of inertia is calculated by integrating the product of the density function, the squared distance from the axis of rotation, and the volume element over the entire volume of the object.

## What is the importance of calculating the moment of inertia?

The moment of inertia is important because it helps us understand how difficult it is to change the rotational motion of an object. It is a crucial factor in designing structures, machines, and vehicles.

## What are some real-world applications of the triple integral for moment of inertia?

The triple integral for moment of inertia is used in various fields, including engineering, physics, and biomechanics. It is used to calculate the stability of buildings, the behavior of rotating machinery, and the movement of human joints and limbs.

## Are there any limitations to using the triple integral for moment of inertia?

While the triple integral is a powerful tool in calculating moment of inertia, it has limitations. It can only be used for rigid bodies with a continuous and uniform mass distribution. It also assumes the object is rotating around a fixed axis.

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