Converting Rectangular to Spherical Iterated Integrals | Help Needed

In summary, converting a rectangular iterated integral to a spherical one involves changing coordinates from rectangular to spherical and adjusting the limits of integration accordingly. It may require using trigonometric identities and some geometry, but visualizing the problem and breaking it down into smaller steps can help in figuring it out.
  • #1
Tater
10
0
Hi everyone,

I am REALLY confused and lost on where about to begin a conversion of a rectangular iterated integral to a spherical iterated integral.


Can someone kind of guide me through on what to do first? Like for example, I drew the initial iterated integral in both 2d and 3d diagrams, but I have no clue what to do from here. The period is about the only thing I can figure out. Can anyone guide me as to what I should be looking for or doing first? I am having a rough time getting started.


Any help is greatly appreciated!
 
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  • #2


Hi there,

First of all, don't worry - converting a rectangular iterated integral to a spherical one can be confusing, but with some guidance, you'll be able to figure it out. Let's start by understanding what an iterated integral is. An iterated integral is a type of multiple integral, where we integrate a function over a specific region in multiple steps. In the case of a rectangular iterated integral, we are integrating over a rectangular region in the xy-plane.

To convert this to a spherical iterated integral, we need to change our coordinates from rectangular (x,y) to spherical (r,θ,φ). This means that we need to express our function in terms of r,θ, and φ and also change our limits of integration to correspond with these new coordinates.

To get started, you mentioned that you have already drawn the initial iterated integral in both 2D and 3D diagrams. This is a great first step as it can help you visualize the problem. Now, let's look at the function itself. Is it already expressed in terms of r,θ, and φ? If not, you'll need to use some trigonometric identities to rewrite it in terms of these variables.

Next, we need to change our limits of integration. In spherical coordinates, r represents the distance from the origin, θ represents the angle from the positive z-axis, and φ represents the angle from the positive x-axis in the xy-plane. So, you'll need to figure out how these new coordinates correspond to the rectangular region you were originally integrating over. This may require some geometry and trigonometry, but once you have the new limits, you can substitute them into your integral.

I hope this helps get you started. Remember, it's always helpful to visualize the problem and break it down into smaller steps. Let me know if you have any other questions or need further guidance. Good luck!
 

1. What is the process for converting rectangular to spherical iterated integrals?

The process for converting rectangular to spherical iterated integrals involves first understanding the differences between the two coordinate systems. Rectangular coordinates use the x, y, and z axes, while spherical coordinates use the radius, inclination, and azimuth angles. Then, the conversion can be done using the appropriate formulas and adjusting the limits of integration accordingly.

2. Why would someone need to convert rectangular to spherical iterated integrals?

Converting between coordinate systems is often necessary in mathematics and science, as different problems may be better suited for different coordinate systems. In the case of integrals, certain problems may be easier to solve using spherical coordinates, so converting from rectangular to spherical can make the problem more manageable.

3. What are the formulas for converting rectangular to spherical coordinates?

The formula for converting from rectangular to spherical coordinates is as follows:
r = √(x^2 + y^2 + z^2)
θ = arctan(y/x)
φ = arccos(z/√(x^2 + y^2 + z^2))

4. Can you provide an example of converting a rectangular integral to a spherical integral?

Sure, let's say we have the integral ∫∫∫ f(x,y,z) dV, where the limits of integration are x from 0 to 2, y from 0 to 2, and z from 0 to 2. To convert this to spherical coordinates, we would use the following limits for the new integral:
r from 0 to 2
θ from 0 to π/4
φ from 0 to π/4
We would also need to convert the integrand f(x,y,z) to f(r,θ,φ) using the formulas mentioned in the previous question.

5. Are there any common mistakes to avoid when converting rectangular to spherical iterated integrals?

Yes, there are a few common mistakes to avoid. One is forgetting to convert the integrand to the new coordinate system. Another is using the wrong limits of integration for the new integral. It is also important to pay attention to the order in which the variables appear in the new integral, as this can affect the final result. Lastly, it's always a good idea to double-check the conversion formulas to make sure they are being applied correctly.

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