Evaluating Triple Integral of G: xyz dV

In summary, to evaluate the given triple integral over the region G bounded by x=1, y=x, y=0, z=0, z=2, one can first evaluate the integral in the triangle 0<=x<=1 and 0<=y<=x while assuming z is constant. This result can then be integrated with z as a variable from 0 to 2. The final answer is 1/3.
  • #1
squenshl
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Given the triple integral [tex]\int\int\int_{G}[/tex] xyz dV
Where G is the region bounded by x=1, y=x, y=0, z=0, z=2.
How do I evaluate it.
Please help.
 
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  • #2
First of all, is there any other constraint on x? I'll assume that x>=0.
Do you know how to change triple integrals into single-variable integrals?
The general idea is to first evaluate the integral while pretending that 2 of the variables are constant. Then you use that result to integrate over the other variables.

In this case it might be easiest to first evaluate the integral in the triangle 0<=x<=1 and
0<=y<=x while assuming that z is constant. Then integrate that result with z as a variable from 0 to 2. The triangle can be evaluated in a similar way. In other words:
[tex]I = \int^{2}_{0}(\int^{1}_{0}(\int^{1-x}_{0} xyz dy)dx)dz[/tex]
 
  • #3
Cheers. I got an answer of 1/3
 

FAQ: Evaluating Triple Integral of G: xyz dV

1. What is a triple integral?

A triple integral is a mathematical concept used in multivariable calculus to evaluate the volume of a three-dimensional region bounded by a function or surface. It involves integrating a function over a given region in three-dimensional space.

2. How is a triple integral different from a regular integral?

A regular integral is a single integral that involves integrating a function over a one-dimensional interval. A triple integral, on the other hand, involves integrating a function over a three-dimensional region, resulting in a volume rather than a single value.

3. What does the "xyz" in the triple integral notation represent?

The "xyz" in the notation represents the three variables over which the function is being integrated. In this case, the function is being integrated over the three variables x, y, and z to find the volume of the given region.

4. How do you evaluate a triple integral?

Evaluating a triple integral involves breaking down the given region into smaller, simpler regions and setting up the integral accordingly. It is then solved using techniques such as Fubini's theorem, where the integral is split into three separate integrals and evaluated using standard integration techniques.

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

Triple integrals have various applications in physics, engineering, and other fields. They can be used to calculate the mass, center of mass, and moment of inertia of three-dimensional objects. They are also used in fluid mechanics to calculate the flow of fluids through three-dimensional regions, and in electromagnetism to calculate the electric and magnetic fields in three-dimensional space.

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