Converting into polar integrals from rectangular

In summary, to convert a rectangular integral into a polar integral, the limits of integration must be expressed in terms of polar coordinates and the integrand must be converted using the formula r = √(x² + y²). This conversion can simplify the integration process and make it easier to integrate functions that are more naturally expressed in polar coordinates. The main difference between rectangular and polar integrals is the coordinate system used, and not all rectangular integrals can be converted into polar integrals. Converting into polar integrals may also have disadvantages, such as requiring extra steps and complexity, and some functions may not have a simple polar form.
  • #1
VinnyCee
489
0
Here is the problem:

Convert [tex]\int_{-1}^{1}\int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}}\;\ln\left(x^2\;+\;y^2\;+\;1\right)\;dx\;dy[/tex] into polar coordinates.

Here is what I have:

[tex]\int_{0}^{2\pi}\int_{0}^{1}\;r\;\ln\left(r^2\;+\;1\right)\;dr\;d\theta[/tex]

Is that the correct conversion? I could list all of the steps that I did to get to that answer, but that would take forever! Can someone check please?
 
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  • #2
That's correct.
 
  • #3


Yes, your conversion is correct! To double check, you can plug in values for r and theta to see if they match up with the original rectangular coordinates. For example, when r = 1 and theta = 0, you get x = 1 and y = 0, which matches with the lower limit of x in the original integral. Great job!
 

1. How do you convert a rectangular integral into a polar integral?

To convert a rectangular integral into a polar integral, you first need to express the limits of integration in terms of polar coordinates. This can be done by substituting x and y with their respective polar equivalents, rcosθ and rsinθ. Then, you can convert the integrand by using the conversion formula r = √(x² + y²).

2. Why is it useful to convert into polar integrals from rectangular?

Converting into polar integrals from rectangular can often simplify the integration process. It can also make it easier to integrate functions that are more naturally expressed in polar coordinates, such as polar curves or regions with circular symmetry.

3. What is the difference between a rectangular integral and a polar integral?

The main difference between a rectangular integral and a polar integral is the coordinate system used for integration. Rectangular integrals use the x and y coordinates, while polar integrals use the r and θ coordinates. Additionally, the limits of integration and the integrand may also be expressed differently in polar form.

4. Can any rectangular integral be converted into a polar integral?

No, not all rectangular integrals can be converted into polar integrals. Some integrals may have regions that are not easily expressed in polar coordinates, or the conversion may result in a more complicated integrand. It is important to consider the integrand and the region of integration before attempting to convert.

5. Are there any disadvantages to converting into polar integrals from rectangular?

One potential disadvantage of converting into polar integrals from rectangular is that it may require extra steps and may be more complicated compared to integrating in rectangular form. Additionally, some functions may not have a simple polar form, making the conversion difficult or impossible.

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