# Help Integrating Tripple Integral for x+y+z=1

• Alem2000
In summary, a triple integral is a mathematical operation used to calculate the volume of a three-dimensional object. It involves setting up limits of integration and using appropriate methods to solve the integral. Its purpose is to find the volume or quantity within a given region in space. Triple integrals have various real-world applications, such as determining mass, center of mass, and electric or magnetic fields.
Alem2000
the question is "evaluate $$\iiint z \,dv$$, of a solid tetrahedron bounded by the four planes x=0,y=0,z=0, and x+y+z=1"
I can set up the problem correctly but i can't seem to integrate it right
$$\int_{0}^1 \int_{0}^{1-x} \int_{0}^{1-x-y} z dzdydx$$
$$(1/2) \int_{0}^1 \int_{0}^{1-x} (1-x-y)^2dydx$$
can somone please show me the last few steps of this problem?

Open the brackets and integrate each term with corresponding limits...

Daniel.

For the evaluation of the inner integral, use the y-anti-derivative :
$$-\frac{1}{3}(1-x-y)^{3}$$

## 1. What is a triple integral?

A triple integral is a type of mathematical operation that involves calculating the volume of a three-dimensional object. It is similar to a regular integral, but instead of finding the area under a curve in two dimensions, it finds the volume under a surface in three dimensions.

## 2. How do you integrate a triple integral?

To integrate a triple integral, you first need to set up the limits of integration for each variable (x, y, and z) and then solve the integral using the appropriate integration method, such as iterated integration or the use of triple integrals in spherical or cylindrical coordinates.

## 3. What is the purpose of integrating a triple integral?

The purpose of integrating a triple integral is to find the volume of a three-dimensional object or the amount of a three-dimensional quantity, such as mass or charge, within a given region in space.

## 4. How do you apply a triple integral to the equation x+y+z=1?

To apply a triple integral to the equation x+y+z=1, you would first need to convert the equation into a form that is suitable for integration. This could involve changing the variables to spherical or cylindrical coordinates, or using other methods such as integration by parts or substitution. Then, you would set up the limits of integration for each variable and solve the integral to find the volume under the surface defined by the equation.

## 5. What are some real-world applications of triple integrals?

Triple integrals have many real-world applications, including calculating the mass or density of objects, finding the center of mass of a three-dimensional object, and determining the electric or magnetic fields in a given region in space.

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