# Triple Integral of xy over a Solid Tetrahedron | Homework Statement

• Larrytsai
In summary, the conversation discusses how to evaluate the triple integral of xy*dxdydz where E is a solid tetrahedron with specific vertices. The person suggests solving it via dzdydx and determining the equation of the plane that is the "top" of the tetrahedron. The integral can then be represented as V=\int_0^{10}\int_0^{f(x)}\int_{0}^{g(x,y)} xy dzdydx.
Larrytsai

## Homework Statement

Evaluate the triple integral of xy*dxdydz where E is the solid tetrahedon with vertices (0,0,0) (10,0,0) (0,8,0) (0,0,5).

## The Attempt at a Solution

Im trying to integrate dx and dy with bounds from 0 to the line that describes them with respect to the z axis,

so for

dy i have the bounds as

0 to y= -8z/5 + 8

dx

0 to x= -2z + 10

and dz

0 to 5

will this work?

I don't think so. I'll give you a suggestion but you may not like it: first solve it via dzdydx because that's more natural and helps you understand the process then you can use what you learn to solve it via dxdydz. But first, you'd have to determine the equation of the plane that is the "top" of the tetrahedron (see below) and that involves computing the cross product to obtain the normal to the plane. Once you do that, then the integral is easy and can be represented as:$$V=\int_0^{10}\int_0^{f(x)}\int_{0}^{g(x,y)} xy dzdydx$$

#### Attachments

• mytet.jpg
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## 1. What is a triple integral?

A triple integral is a type of mathematical operation used to calculate the volume under a three-dimensional surface. It involves integrating a function over a solid region in three dimensions.

## 2. What is the meaning of "xy" in the triple integral of xy over a solid tetrahedron?

In this context, "xy" refers to a function of two variables, x and y. The triple integral is calculated by integrating this function over the solid tetrahedron.

## 3. How is a solid tetrahedron defined in this context?

A solid tetrahedron is a three-dimensional shape with four triangular faces. In the context of a triple integral, it is defined as a region in three-dimensional space bounded by the xy-plane and four triangular planes.

## 4. How do you set up the limits for the triple integral of xy over a solid tetrahedron?

The limits for a triple integral are determined by the boundaries of the solid region being integrated over. In the case of a solid tetrahedron, the limits for x, y, and z will depend on the equations of the four triangular planes that define the tetrahedron.

## 5. What is the importance of calculating a triple integral over a solid tetrahedron?

Triple integrals are useful in many fields of science, particularly in physics and engineering, where they are used to calculate volume, mass, and other physical quantities. In the case of a solid tetrahedron, the triple integral allows us to find the volume under a three-dimensional surface within the boundaries of the tetrahedron.

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