What is the best approach for solving a tricky triple integral problem?

In summary, this conversation discusses a triple integral with given bounds and a function of x, y, z. The region of integration is a half-circle in the xz plane and attempts at solving the integral through integration by parts have been unsuccessful. The suggestion is made to try a transformation from rectangular coordinates to something else.
  • #1
halcyone
2
0

Homework Statement



A triple integral, with the bounds, from outer to inner:
integrate from -1 to 1 with respect to x
integrate from 0 to 1-x^2 with respect to y
integrate from 0 sqrt (y) with respect to z
on the function x^2*y^2*z^2

Homework Equations


none

The Attempt at a Solution


I know what kind of a region it is. The region intersects at x^2+z^2=1. However, my attempts at solving this integral lead to a messy, impossible looking integral, and I am fairly sure that this integral requires no more than integration by parts. I've tried changing the bounds, such as letting D=half-circle in xz plane, and let the bounds on y be from z^2 to 0, but they lead to similar problems. What else can I do?...
 
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  • #2
Try a transformation from rectangular to something else...
 

FAQ: What is the best approach for solving a tricky triple integral problem?

What is a triple integral?

A triple integral is a mathematical concept that extends the idea of a regular integral to three-dimensional space. It involves finding the volume under a three-dimensional function or surface.

How is a triple integral different from a regular integral?

A regular integral finds the area under a curve in two-dimensional space, while a triple integral finds the volume under a surface in three-dimensional space.

What are the applications of triple integrals?

Triple integrals have various applications in physics, engineering, and other fields. They can be used to calculate volumes, mass, center of mass, moments of inertia, and other physical quantities in three-dimensional space.

What is the process for solving a triple integral problem?

The process for solving a triple integral involves breaking down the problem into smaller parts, using appropriate integration techniques, and evaluating the integral using limits of integration for each variable.

What are some common techniques for evaluating triple integrals?

Some common techniques for evaluating triple integrals include using the order of integration, changing the variable of integration, and using symmetry to simplify the integral. Other techniques such as cylindrical or spherical coordinates may also be useful depending on the problem.

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