Consider the tetrahedron 𝑇
, with vertices (0,0,0),(1,0,0),(1,4,0)
and (1,0,1)
. Express the following triple integral as an iterated integral:

To express the given triple integral over the tetrahedron \( T \) as an iterated integral, we first describe the region \( T \) in terms of the variables \( x \), \( y \), and \( z \). The vertices of the tetrahedron are:

- \( A = (0,0,0) \)
- \( B = (1,0,0) \)
- \( C = (1,4,0) \)
- \( D = (1,0,1) \)

We will use these vertices to set up the bounds for the iterated integral. The plane containing the base triangle \( \triangle ABC \) is \( z = 0 \), while the plane containing the top triangle \( \triangle ABD \) will provide the upper bound for \( z \). We will first describe the projection of the tetrahedron onto the \( xy \)-plane.

### 1. Boundaries for \( x \) and \( y \)
In the \( xy \)-plane, the projection is a triangle with vertices at \( (0,0) \), \( (1,0) \), and \( (1,4) \).

For \( 0 \leq x \leq 1 \), the boundary for \( y \) is given by:
- \( y = 0 \) (from \( (0,0) \) to \( (1,0) \))
- \( y = 4(x - 1) \) (from \( (1,0) \) to \( (1,4) \))

Thus, \( y \) ranges from 0 to \( 4(1 - x) \).

### 2. Boundaries for \( z \)
For a fixed point \( (x, y) \), the value of \( z \) ranges from 0 to the plane determined by the points \( (1,0,0) \), \( (1,4,0) \), and \( (1,0,1) \). The equation of this plane can be determined as follows:

Since three points on the plane are known, the equation of the plane can be derived using the determinant approach. However, noticing that the plane intersects the \( z \)-axis at 1 and does not depend on \( x \), we can directly express \( z \) as:

\( z = \frac{1 - y}{4} \).

Thus, \( z \) ranges from 0 to \( \frac{1 - y}{4} \).

### Iterated Integral Setup
The region \( T \) can be described with the bounds found above:

\[
\int_{0}^{1} \int_{0}^{4(1-x)} \int_{0}^{\frac{1-y}{4}} dz \, dy \, dx
\]

Here, the integrand is assumed to be 1 (for the volume) unless otherwise specified.

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Would you like a detailed step-by-step explanation or have any other questions?  

Here are some related questions you might find useful:

1. What is the geometric interpretation of the triple integral?
2. How do you derive the equation of a plane from three points?
3. What are other methods to describe a region for triple integrals?
4. How do you find the volume of a tetrahedron using determinants?
5. What are the alternative coordinate systems for triple integrals?
6. Can you calculate the volume using different orders of integration?
7. How do changes in the order of integration affect the integral limits?
8. What is the Jacobian, and how is it used in triple integrals?

**Tip:** When setting up a triple integral, carefully determine the projection of the region onto a plane, as it simplifies finding the bounds for each variable.

Learn how to express a triple integral over a tetrahedron T as an iterated integral. Understand the geometric interpretation, projection onto the xy-plane, and setup of integral bounds for x, y, and z variables. Suitable for undergraduate mathematics students.

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