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Tropicalism
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1. Evaluate the double integral ∫∫arctan(y/x) dA by converting to polar coordinates over the Region R= { (x,y) | 1≤x^2+y^2≤4 , 0≤y≤x }
My attempt at solving
Converting to polar using x=rcosθ and y=rsinθ I get
∫∫arctan(tan(θ))r drdθ
I understand that I have to integrate first with respect to r and then with respect to θ but I'm not sure what the region is supposed to look like over which I am integrating so I don't know what my bounds should be. I understand that 1≤x^2+y^2≤4 is basically 2 circles, one with radius 1, and one with radius 2 and my region is between them. I don't know how 0≤y≤x affects it though, I thought it would just make it the top right portion of the circle but that's incorrect.
Any help is greatly appreciated, thanks.
My attempt at solving
Converting to polar using x=rcosθ and y=rsinθ I get
∫∫arctan(tan(θ))r drdθ
I understand that I have to integrate first with respect to r and then with respect to θ but I'm not sure what the region is supposed to look like over which I am integrating so I don't know what my bounds should be. I understand that 1≤x^2+y^2≤4 is basically 2 circles, one with radius 1, and one with radius 2 and my region is between them. I don't know how 0≤y≤x affects it though, I thought it would just make it the top right portion of the circle but that's incorrect.
Any help is greatly appreciated, thanks.
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