# Evaluate double integral by changing to polar coordinates

• GreenLantern
In summary, the conversation discusses an error made by the speaker in regards to including the bound y<=x in their work. Despite being advised to do so, they were still unable to find their mistake. They also mention the use of arctan and its relationship to tan, as well as the need to split along quadrants. The speaker expresses gratitude for the help provided by the other person.
GreenLantern

what'd I do wrong?
I was told I didn't include the bound y<=x but that still hasn't helped me figure out where I miss stepped

thanks
-Ben

Last edited:
Ok, why did you ignore the advice that you didn't include the bound y<=x? You didn't, so you angular limits are wrong. For another thing, arctan(tan(t)) is not necessarily t. tan(5*pi/4)=1. arctan(1)=pi/4. You'll have to split along quadrants as well.

I didn't ignore the advice, it just wasn't enough for me to go on to find and correct my error.
I understand now. Thank you for your help.
-GL

## 1. How do you convert a double integral to polar coordinates?

To convert a double integral to polar coordinates, you need to substitute the variables x and y with their polar coordinate equivalents: x = rcosθ and y = rsinθ. You also need to replace the dx dy term with r dr dθ. Remember to adjust the limits of integration accordingly.

## 2. Why would you want to use polar coordinates for a double integral?

Polar coordinates are useful for evaluating integrals that have circular or radial symmetry. This can make the integral easier to solve, as the boundaries of the region of integration may be simpler to define in polar coordinates.

## 3. What are the advantages of using polar coordinates over rectangular coordinates for a double integral?

Using polar coordinates can simplify the integrand and make it easier to evaluate, especially for integrals with circular or radial symmetry. It can also help to reduce the number of variables and parameters in the integral.

## 4. How do you determine the new limits of integration when converting to polar coordinates?

The new limits of integration in polar coordinates will depend on the shape and boundaries of the region of integration. To determine the limits, you can draw a diagram and use geometric reasoning to find the relationships between the rectangular and polar coordinates. You can also use trigonometric identities to express the limits in terms of θ.

## 5. Can any double integral be converted to polar coordinates?

No, not all double integrals can be converted to polar coordinates. The integral must have circular or radial symmetry in order for the conversion to be valid. Additionally, the region of integration must be more easily defined in polar coordinates.

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