**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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