(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate [tex]\iiint_\textrm{V} |xyz|dxdydz[/tex]

where [tex]V = \{(x,y,z) \in ℝ^3:\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ≤ 1\}[/tex]

2. Relevant equations

Change of Variables Theorem:

[tex]\int_\textrm{ψ(u)} f(x)dx = \int_\textrm{K} f(\Psi(u))|detD\Psi(u)|du[/tex]

Examples:

1)

For a ball of radius a,

[tex]B(a) = \{(x,y,z) \in ℝ^3:x^2 + y^2 + z^2 ≤ a^2\}[/tex]

[tex]vol B(a) = \int_\textrm{B(a)}1 dxdydz[/tex]

[tex]= \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{a} \rho^2 sin \phi d \rho d \phi d \theta[/tex](change of variables to spherical coodinates)

2)

For a continuous function f: D → ℝ where

[tex]D = \{(x,y) :\frac{x^2}{a^2} + \frac{y^2}{b^2} ≤ 1\}[/tex]

define ψ: ℝ^2 → ℝ^2 by ψ(au, bv) for all u,v in ℝ^2. ψ is a smooth change of variables.

Then

[tex]\int_\textrm{D}f(x,y)dxdy = ab\int_\textrm{u^2 + v^2 ≤ 1}f(au, bv) du dv[/tex]

[tex] = ab\int_{0}^{2\pi} \int_{0}^{1}f(ar cos \theta, br sin \theta) r dr d \theta . [/tex]

(change of variables to polar coordinates)

3. The attempt at a solution

Based on those examples above in the book, I set this up:

Let ψ(u,v,w) = (au, bv, cw)

Then

[tex]\iiint_\textrm{V} |xyz|dxdydz = abc\int_\textrm{u^2 + v^2 + w^2≤ 1}|abcuvw| du dvdw[/tex]

change of variables to spherical coordinates:

[tex] = abc\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{1} |abc\rho^3 cos \phi sin^2 \phi cos \theta sin \theta|\rho^2 sin \phi d \rho d \phi d \theta . [/tex]

[tex] = (abc)^2\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{1} \rho^5 cos \phi sin^3 \phi cos \theta sin \theta d \rho d \phi d \theta . [/tex]

p^3 is positive on [0, 1] so I ignored the absolute value lines.

This eventually led to an answer of 0 since one of the antiderivates is sin^4(phi)on [0,pi] which is zero. This is wrong. So I'm guessing the integral I set up or the way I evaluated it is wrong. But I got half credit for it, so I assume some part of it is right.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integral calculus involving Change of Variables Theorem

**Physics Forums | Science Articles, Homework Help, Discussion**