Triple integral over a sphere in rectangular coordinates

In summary, the conversation discusses the evaluation of a triple integral with boundaries given by a sphere in the first octant with radius 2, using rectangular, spherical, and cylindrical coordinates. The attempt at a solution involves transformations and a Jacobian calculation, but the conversation leads to a revised solution with bounds that depend on the previous variable. The final integral is over a quarter circle in the xy-plane, with x ranging from 0 to 2 and y ranging from 0 to the square root of 4 minus x squared.
  • #1
Batmaniac
24
0

Homework Statement



Evaluate the following integral:

[tex]
\iiint \,x\,y\,z\,dV
[/tex]

Where the boundaries are given by a sphere in the first octant with radius 2.

The question asks for this to be done using rectangular, spherical, and cylindrical coordinates.

I did this fairly easily in spherical and rectangular coordinates, except for the fact that I got two different answers and I can't figure out where I went wrong! That's not a problem though because I can fix that.


The Attempt at a Solution



How would I do this problem in rectangular coordinates? My integral would look like this:

[tex]
\int_{0}^{{\sqrt{4-x^2-y^2}}}\int_{0}^{{\sqrt{4-x^2-z^2}}}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx
[/tex]

Which, without some clever transformations and an extremely messy Jacobian calculation, looks unsolvable.
 
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  • #2
I think you need to rethink your bounds on that one...
 
  • #3
How does this look then?

[tex]
\int_{0}^{2}\int_{0}^{2}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx
[/tex]
 
  • #4
Hmm, MATLAB tells me that's zero.
 
  • #5
Your dy limit should depend on x.
 
  • #6
Batmaniac said:
How does this look then?

[tex]
\int_{0}^{2}\int_{0}^{2}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx
[/tex]

That would be over a square in the xy-plane rising up to the sphere.

Projecting the sphere into the xy-plane gives you the quarter circle x2+ y2= 4, with [itex]0\le x\le 2[/itex], [itex]0\le y\le 2[/itex]. You can let x go from 0 to 2 but then, for each x, y ranges from 0 to [itex]\sqrt{4- x^2}[/itex].
 

1. What is a triple integral over a sphere in rectangular coordinates?

A triple integral over a sphere in rectangular coordinates is a mathematical operation that calculates the volume of a three-dimensional shape called a sphere using rectangular coordinates. It involves integrating a function over a spherical region in three-dimensional space.

2. How is a triple integral over a sphere in rectangular coordinates different from a regular triple integral?

A triple integral over a sphere in rectangular coordinates differs from a regular triple integral because the region of integration is a three-dimensional sphere rather than a rectangular box. This requires a change in the limits of integration and the use of spherical coordinates instead of rectangular coordinates.

3. What is the formula for calculating a triple integral over a sphere in rectangular coordinates?

The formula for calculating a triple integral over a sphere in rectangular coordinates is ∭f(x,y,z) dV = ∫∫∫f(x,y,z) dx dy dz, where the limits of integration are determined by the spherical region and the function f(x,y,z) represents the function being integrated.

4. What are some applications of triple integrals over a sphere in rectangular coordinates in science?

Triple integrals over a sphere in rectangular coordinates have various applications in science, such as calculating the mass or volume of objects with spherical symmetry, determining electric and magnetic fields around charged or magnetic spheres, and solving certain differential equations in physics and engineering.

5. Are there any limitations to using triple integrals over a sphere in rectangular coordinates?

While triple integrals over a sphere in rectangular coordinates have many applications, they are limited in their ability to accurately calculate the volume of complex shapes. In some cases, other methods such as numerical integration may be more suitable for solving these types of problems.

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