Selecting the correct bounds for polar integrals

In summary, the given double integral of arctan(y/x) over the region R: 1≤x2+y2≤4, 0≤y≤x has bounds of r from 1 to 2 and θ from 0 to π/4 in polar coordinates. The integration is straightforward, but assistance is needed with determining the bounds.
  • #1
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Hi!

Here's a question I am working on:

Double integral of arctan(y/x).

where R: 1≤x2+y2≤4, 0≤y≤x.

I have the bounds for r as 1 to 2, but for θ I don't know if I should use ∏/4 to ∏/2 or 0 to ∏/2. How do I know which one?

The integration is easy, but I need help with the bounds.

Thanks.
 
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  • #2
My point of view :
[ 0 < y < x ] means that y and x are more then zero, [ y < x ] means that all points are below/under the line [ y = x ], line [ y = x ] in the polar coordinate system has an equation :: [ angle = pi / 4 ]. Then the bounds in polar system from 0 to pi/4.
arctg (y / x) = ( r*sin( ang ) / [ r*cos( ang ) ] ) = acrtg( tg (ang) ) = ang
Integral ( angle * R * ( d R ) * ( d angle ) ) = 3*pi*pi / 64
 

1. What are the types of bounds for polar integrals?

The most common types of bounds for polar integrals are rectangular bounds and circular bounds. Rectangular bounds are defined by a range of values for the radius and an angle, while circular bounds are defined by a radius and two angles.

2. How do I determine the correct bounds for a polar integral?

The correct bounds for a polar integral depend on the shape and orientation of the region being integrated. One method is to graph the polar equation and identify the limits of the region. Another method is to use the given bounds for the cartesian coordinates and convert them to polar form.

3. Can I use negative values for the bounds in polar integrals?

Yes, negative values can be used for the bounds in polar integrals. Negative values for the radius will simply result in integration over the opposite direction or quadrant.

4. How do I change the order of integration for polar integrals?

The order of integration for polar integrals can be changed by swapping the bounds for the radius and angle. This will result in a different orientation of the region being integrated.

5. Are there any special cases for selecting bounds in polar integrals?

Yes, there are special cases where the bounds may need to be adjusted. This includes when the region being integrated is not symmetric, or when the polar equation includes negative values for the radius, which may require a change in the order of integration. It is important to carefully analyze the region and equation to determine the appropriate bounds.

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