**Triple Integral in a Tetrahedron**Consider the tetrahedron $ T $ with vertices at the points $ (3, 0, 0) $, $ (0, 4, 0) $, $ (0, 0, 5) $, and the origin $(0, 0, 0)$. To express the triple integral of a function $f(x, y, z)$ over this tetrahedron $ T $, we aim to write it in the form:\[ \iiint_{T} f(x, y, z) \, dV \]We can set up the following iterated integral:\[ \int_{0}^{3} \int_{0}^{u(x)} \int_{0}^{v(x, z)} f(x, y, z) \, dy \, dz \, dx \]where $u(x)$ and $v(x, z)$ are the upper bounds of $z$ and $y$ respectively, as functions of $x$ and $z$.Determine the functions $u(x)$ and $v(x, z)$:1. **Finding $ u(x) $:** $u(x)$ denotes the upper bound for $ z $ in terms of $ x $. It represents the plane equation intersecting with the $ z $-axis. 2. **Finding $ v(x, z) $:** $v(x, z)$ defines the upper bound for $ y $ in terms of $ x $ and $ z $, given the constraints of the tetrahedral region. Based on the given vertices, calculations for these bounds will result in:\[ u(x) = 5 - \left(\frac{5}{3}\right)x \]\[ v(x, z) = 4 - \left(\frac{4}{3}\right)x - \left(\frac{4}{5}\right)z \]Thus:\[ u(x) = 5 - \frac{5}{3}x \]\[ v(x, z) = 4 - \frac{4}{3}x - \frac{4}{5}z \]Incorporate these bounds into the iterated integral:\[ \int_{0}^{3} \int_{0}^{5 - \frac{5}{3}x} \int_{0

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Answered: Let T be the tetrahedron with vertices…

Let T be the tetrahedron with vertices (3, 0, 0), (0, 4, 0), (0, 0, 5) and the origin. Write f(x, y, z) dV in the form 3 pu(z) pv(z,z) I"" f(x, y, z) dy dz dæ u(x) = v(x, z) =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Triple Integral in a Tetrahedron**

Consider the tetrahedron $ T $ with vertices at the points $ (3, 0, 0) $, $ (0, 4, 0) $, $ (0, 0, 5) $, and the origin $(0, 0, 0)$. To express the triple integral of a function $f(x, y, z)$ over this tetrahedron $ T $, we aim to write it in the form:

\[ \iiint_{T} f(x, y, z) \, dV \]

We can set up the following iterated integral:

\[ \int_{0}^{3} \int_{0}^{u(x)} \int_{0}^{v(x, z)} f(x, y, z) \, dy \, dz \, dx \]

where $u(x)$ and $v(x, z)$ are the upper bounds of $z$ and $y$ respectively, as functions of $x$ and $z$.

Determine the functions $u(x)$ and $v(x, z)$:

1. **Finding $ u(x) $:**

$u(x)$ denotes the upper bound for $ z $ in terms of $ x $. It represents the plane equation intersecting with the $ z $-axis.

2. **Finding $ v(x, z) $:**

$v(x, z)$ defines the upper bound for $ y $ in terms of $ x $ and $ z $, given the constraints of the tetrahedral region.

Based on the given vertices, calculations for these bounds will result in:

\[ u(x) = 5 - \left(\frac{5}{3}\right)x \]

\[ v(x, z) = 4 - \left(\frac{4}{3}\right)x - \left(\frac{4}{5}\right)z \]

Thus:

\[ u(x) = 5 - \frac{5}{3}x \]
\[ v(x, z) = 4 - \frac{4}{3}x - \frac{4}{5}z \]

Incorporate these bounds into the iterated integral:

\[ \int_{0}^{3} \int_{0}^{5 - \frac{5}{3}x} \int_{0

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