Evaluate the triple integral for specified function

In summary: G8gZG8geW91ciBleHByZXNzIHlvdXIgYm91bmRzIHJlYWR5Lg==In summary, the problem requires evaluating the triple integral of the function f(x,y,z) = xey-2z over the specified box B, with bounds 0<x<2, 0<y, z>1. There may be some confusion with the bounds for y and z, but it is possible to set up the integral by understanding that 0<=y<infinity and 1<=z<infinity. The region for this problem is a rectangular prism, and using the correct bounds will lead to the correct solution.
  • #1
musicmar
100
0

Homework Statement


Evaluate the triple integral for specified function and box B.

f(x,y,z) = x ey-2z
0<x<2, 0<y, z>1
(The < and >'s should be less than or equal to but, I don't know how to write that here)


Homework Equations





The Attempt at a Solution



I know how to evaluate iterated integrals, but I am confused by the bounds in this problem. y and z only have one bound, so I don't know how to set up the integral.
 
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  • #2
musicmar said:
y and z only have one bound, so I don't know how to set up the integral.


Why don't they just go to infinity then? Why not just take a stab at it and say, let's integrate dzdydx and try anyway to set up reasonable limits? If z>1 then it's limits should be (1,infty}, same dif with y, (0,infty), and x goes from 0 to 2. Here goes:

[tex]\int_0^2 \int_0^{\infty}\int_1^{\infty} xe^{y-2z} dzdydx[/tex]

Now, I haven't tried to integrate that. Maybe when we do, we'll get into a tough integrand to handle. Then may want to consider a different integration order if that would make the integration easier.
 
  • #3
musicmar said:

Homework Statement


Evaluate the triple integral for specified function and box B.

f(x,y,z) = x ey-2z
0<x<2, 0<y, z>1
(The < and >'s should be less than or equal to but, I don't know how to write that here)
Use <= for less than or equal to, and >= for greater than or equal to.
musicmar said:

Homework Equations





The Attempt at a Solution



I know how to evaluate iterated integrals, but I am confused by the bounds in this problem. y and z only have one bound, so I don't know how to set up the integral.
There are implied bound for y and z.

0 <= y < infinity
1 <= z < infinity

Do you know what the region in this problem looks like?
 
  • #4
Thank you. Is it a rectangular prism? y >= 0 is a plane bounded by x, and z would provide the height. But does this help me integrate it?
 
  • #6
I had to work the same problem. I have found discrepancies in the book before regarding notation, so when I couldn't solve the problem I assumed the boundary meant 0≤X≤2; 0≤y,z≤1→0≤y≤1; 0≤z≤1. Alternately, the box B=[0,2], [0,1], [0,1]. When I worked the problem with these boundaries I got the same answer as the book.
Hope it helps.
Also, you can get "less than or equal to" by underlining the "less than" symbol.
 
  • #7
musicmar said:

Homework Statement


Evaluate the triple integral for specified function and box B.

f(x,y,z) = x ey-2z
0<x<2, 0<y, z>1
(The < and >'s should be less than or equal to but, I don't know how to write that here)


Homework Equations





The Attempt at a Solution



I know how to evaluate iterated integrals, but I am confused by the bounds in this problem. y and z only have one bound, so I don't know how to set up the integral.

You can write <= and >=, or else you can click on the "Quick symbols" from the panel at the side of the input window, to get ≤ and ≥.

RGV
 

Related to Evaluate the triple integral for specified function

1. What is a triple integral?

A triple integral is an extension of a regular integral, where instead of integrating over a single variable, we integrate over three variables. It involves finding the volume under a three-dimensional surface or region.

2. How do you set up a triple integral?

A triple integral is set up using the Cartesian coordinates (x, y, z) and involves three nested integrals, each representing the integration over one of the variables. The innermost integral is typically with respect to z, the middle integral is with respect to y, and the outermost integral is with respect to x.

3. What is the order of integration in a triple integral?

The order of integration in a triple integral can vary, but it is typically written as dzdydx. This means that the innermost integral is with respect to z, the middle integral is with respect to y, and the outermost integral is with respect to x. However, the order can be changed depending on the function and the region of integration.

4. How do you evaluate a triple integral?

To evaluate a triple integral, you need to first set up the integral using the appropriate limits of integration. Then, you can use techniques such as substitution, integration by parts, or partial fractions to simplify the integrand. Finally, you can use numerical methods or integration tables to evaluate the integral.

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

Triple integrals have various applications in physics, engineering, and other fields. For example, they can be used to calculate the mass, center of mass, and moments of inertia of three-dimensional objects. They are also used in calculating electric and gravitational potentials, heat transfer, and fluid mechanics problems.

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