Calculating Triple Integrals in Mathematica

In summary, the conversation is about evaluating a triple integral with cylindrical coordinates and the problem is provided by the instructor. There is confusion about the source and purpose of the integral, as well as potential issues with the limits of integration.
  • #1
november1992
120
0

Homework Statement



Evaluate ∫∫∫[itex]\sqrt{x^{2} + y^{2}}[/itex] dA where R is the region bounded by the paraboloid y=x^2+z^2 and the plane y=4

Homework Equations


I believe this is a problem where cylindrical coordinates would be useful

0 ≤ z ≤ [itex]\sqrt{4-x^2}[/itex]
0 ≤ r ≤ 2 ( I think this is wrong).
0 ≤ θ ≤ 2[itex]\pi[/itex]

The Attempt at a Solution



Integrate[ r^2, {\[Theta], 0, 2*Pi}, {r, 0, 2}, {z, 0, Sqrt[4 - r*Cos[\[Theta]]^2]}]

There isn't an output when I enter this line. I checked for the syntax on the wolfram reference page so I think the problem is with the limits of integration.
 
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  • #2
What is the source of your integral? What is it supposed to represent? I say this because you have a triple integral with dA rather than dV.
 
  • #3
This problem was given to me by my instructor.
 
  • #4
Yeah, I figured it wasn't revealed to you in a dream. But what is the ultimate source of this integral? Is it a problem out of a book? Have you solved faster than light travel? Is a new mathematics about to be revealed to us?

You realize, I hope, that there seems to be something missing: You have a triple integral of a function with respect to what appears to be a differential element of area dA. Three integrations, two variables with respect to which the integration could be carried out. It's not clear how this integral is supposed to be calculated.
 
  • #5


I would first like to commend your attempt at using Mathematica to solve this problem. It is always useful to use computational tools to help solve complex integration problems.

Upon reviewing your attempt, I believe the issue lies with the limits of integration. The region R described by the problem is a paraboloid, not a cylinder, so using cylindrical coordinates may not be the most appropriate approach. Instead, I would suggest using spherical coordinates, where the paraboloid can be described by the equation ρ = θ^2 + z^2.

The limits of integration in spherical coordinates would be:

0 ≤ ρ ≤ 2 (since this is the maximum value of θ^2 + z^2 for the paraboloid)
0 ≤ θ ≤ 2π
0 ≤ φ ≤ π/2 (since the plane y=4 bounds the region above the xy-plane)

Using these limits, the triple integral would be:

Integrate[ρ^2*sin(φ), {θ, 0, 2π}, {ρ, 0, 2}, {φ, 0, π/2}]

I hope this helps and provides a solution to your problem. Keep exploring and using computational tools to solve complex problems in science!
 

1. How do I input a triple integral in Mathematica?

To input a triple integral in Mathematica, you can use the Integrate function with three variables and the desired limits of integration. For example, Integrate[f[x,y,z],{x,x1,x2},{y,y1,y2},{z,z1,z2}] would calculate the triple integral of f[x,y,z] over the region defined by the limits x1 to x2 for x, y1 to y2 for y, and z1 to z2 for z.

2. How do I change the order of integration in a triple integral?

To change the order of integration in a triple integral, you can use the Integrate function with the desired order of variables and limits of integration. For example, Integrate[f[x,y,z],{z,z1,z2},{y,y1,y2},{x,x1,x2}] would calculate the triple integral of f[x,y,z] over the region defined by the limits x1 to x2 for x, y1 to y2 for y, and z1 to z2 for z in that order.

3. How do I evaluate a numerical triple integral in Mathematica?

To evaluate a numerical triple integral in Mathematica, you can use the NIntegrate function instead of Integrate. This function requires the same input parameters, but will return a numerical value instead of an exact solution. For example, NIntegrate[f[x,y,z],{x,x1,x2},{y,y1,y2},{z,z1,z2}] would calculate the numerical triple integral of f[x,y,z] over the region defined by the limits x1 to x2 for x, y1 to y2 for y, and z1 to z2 for z.

4. Can I use variables or functions in the limits of integration for a triple integral?

Yes, you can use variables or functions in the limits of integration for a triple integral in Mathematica. This allows you to integrate over more complex regions. Just make sure that the variables or functions are defined and have numerical values before evaluating the integral.

5. Is there a way to visualize the region of integration for a triple integral in Mathematica?

Yes, you can use the RegionPlot3D function to visualize the region of integration for a triple integral in Mathematica. This function will create a three-dimensional plot of the region, which can be useful for understanding the bounds of integration and checking for any errors. Just make sure to input the same limits of integration as you would for the Integrate or NIntegrate functions.

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