Converting triple integral coordinates

In summary, the conversation discusses setting up a triple integral in rectangular, cylindrical, and spherical coordinates for a solid sphere of radius R. The bounds and integrands for each coordinate system are provided and confirmed by the expert. The conversation ends with a welcome to the Physics forum.
  • #1
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1. consider the triple integral (x^2 +Y^2) dV where it is bounded by a solid sphere of radius R. Set up the integral using rectangular coordinatesI tried setting this up with the bounds [ -sqrt(R^2-x^2-Y^2) <= Z <= sqrt(R^2-x^2-Y^2) ,
-R <= X <= R , -sqrt(R^2-x^2) <= Y <= sqrt(R^2-x^2) ] am I on the right path?
 
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  • #2
Yes, but make sure you have them in the correct order when you write down the integrals.
 
  • #3
Fantastic! Yea I know that it goes dz dy dx
now How about if I rewrite to cylindrical coordinates?
0<= theta <= 2pi , 0<=r<=R , -sqrt(R^2-r^2)<=z<= sqrt(R^2-r^2) with the integrand being r^3 dz dr dtheta
 
  • #4
Yes, you've got it.
 
  • #5
first, thanks so much for the help

just to make sure I fully understand all of these triple integrals, to set it up in spherical I should get
0<= theta<= 2pi, 0<= phi <= pi, 0<= p <= R,
where the integrand is (p^4) (sin(FI))^3 dp dphI dtheta
 
  • #6
Yes. Very nice. And welcome to the Physics forum.
 

Related to Converting triple integral coordinates

1. What is the purpose of converting triple integral coordinates?

Converting triple integral coordinates allows us to change the limits of integration and the order of integration, making it easier to evaluate and solve the integral.

2. How do I convert from rectangular to cylindrical coordinates?

To convert from rectangular to cylindrical coordinates, use the equations x = rcos(theta), y = rsin(theta), and z = z, where r is the distance from the origin to the point in the xy-plane, and theta is the angle between the positive x-axis and the line segment connecting the origin to the point.

3. Can I convert from spherical to cylindrical coordinates?

Yes, you can convert from spherical to cylindrical coordinates by using the equations r = rsin(phi), theta = theta, and z = rcos(phi), where r is the distance from the origin to the point, theta is the angle between the positive x-axis and the line segment connecting the origin to the point, and phi is the angle between the positive z-axis and the line segment connecting the origin to the point.

4. What is the correct order of integration for triple integrals?

The correct order of integration for triple integrals is dx dy dz or dz dy dx. This means that the innermost integral is with respect to x, then y, and finally z.

5. How do I determine the limits of integration for a converted triple integral?

The limits of integration for a converted triple integral can be determined by considering the boundaries of the region in each coordinate system and converting them to the new coordinate system. It is important to pay attention to the order of integration to ensure the correct limits are used.

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