Selecting the correct bounds for polar integrals

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SUMMARY

The discussion focuses on determining the correct bounds for polar integrals, specifically for the double integral of arctan(y/x) over the region defined by 1≤x²+y²≤4 and 0≤y≤x. The correct bounds for r are established as 1 to 2, while the bounds for θ should be set from 0 to π/4, as this range accurately represents the area where y is less than x in polar coordinates. The integration process is straightforward once the bounds are correctly defined, leading to the final result of the integral being 3π²/64.

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  • Understanding of polar coordinates and their conversion from Cartesian coordinates
  • Familiarity with double integrals and their applications in calculus
  • Knowledge of trigonometric functions, specifically arctan and its properties
  • Experience with integration techniques in multivariable calculus
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  • Study the conversion between Cartesian and polar coordinates in detail
  • Learn about the properties and applications of double integrals in various contexts
  • Explore the use of arctan in integration and its geometric interpretations
  • Investigate advanced integration techniques, including change of variables in multiple integrals
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Students and professionals in mathematics, particularly those focusing on calculus, multivariable analysis, and anyone involved in solving polar integrals.

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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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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
 

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