# Multivariable Triple Integral - Calculus Physics/Math Problem

• aligator11
In summary, the conversation is about solving a calculus problem involving evaluating the moment of inertia of a homogeneous solid bounded by a specific area. The person has reached a point where they have expanded the formula and are unsure of the next steps to take. They are also seeking clarification on the variables and coordinate system to use.
aligator11
Homework Statement
Evaluate the Moment of Inertia with respect to Oz axis of the [...] solid A: (x^2+y^2+z^2)^2<=z
Relevant Equations
Formula for moment of inertia: Iz=SSS(x^2+y^2)*constant-density*dxdydz

Cylinder Jacobian: SSS(x,y,z)*|r|dθdrdφ

Cylinder Jacobian substitution parameters:
x=rsin(θ)cos(φ)
y=rsin(θ)sin(φ)
z=rcos(θ)
Jacobian -->|r|=r^2*cos(θ)
Hello everybody.

If anyone could help me solve the calculus problem posted below, I would be greatful.

Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A

Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive something like this: r=(cos(θ))^(1/3) for the definite intergral of my radius. I'm not sure what is the next step I should take...

Thank you all the great souls which are able to help me in that matter.

Cheers!

Well, first of all, do you want to integrate using cylindrical coordinates? Or using spherical coordinates? Because you say cylindrical but the variables you define are spherical, so think about this. Also, check the value you get for the Jacobian.

## 1. What is a multivariable triple integral?

A multivariable triple integral is a mathematical concept used to calculate the volume of a three-dimensional region in space. It involves integrating a function over a three-dimensional region, with each variable representing a different dimension.

## 2. How is a multivariable triple integral different from a regular integral?

A multivariable triple integral is different from a regular integral in that it involves integrating a function over a three-dimensional region, rather than just a one-dimensional interval. This means that there are multiple variables involved, and the integration process is more complex.

## 3. What is the purpose of using a multivariable triple integral?

The purpose of using a multivariable triple integral is to calculate the volume of a three-dimensional region in space. This can be useful in various fields such as physics, engineering, and mathematics.

## 4. What are some real-life applications of multivariable triple integrals?

Multivariable triple integrals have many real-life applications, such as calculating the mass of an object with varying density, determining the electric field of a charged object, and finding the center of mass of a three-dimensional object.

## 5. What are some common techniques for solving multivariable triple integrals?

Some common techniques for solving multivariable triple integrals include using the triple integral formula, converting to cylindrical or spherical coordinates, and using symmetry to simplify the integral. It is also helpful to break down the region into smaller, simpler shapes and use the properties of integrals to solve them separately.

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