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# double integral on non-rectangular area

Posted 9 years ago
 How can I do it on Mathematica 9? Ex: integral of sqrt(4-x^2-y^2) dy dx where x^2+y^2-2*x<=0.
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Posted 9 years ago
 Looking at my messy result, it is actually a sum of several integrals which Mathematica returned unevaluated.
Posted 9 years ago
 Doing the same thing numerically (using NIntegrate) gives the result 19.9868 in 0.34 s. (Mathematica 10.0, Windows 8.1). Actually the result has a tiny imaginary part 5.5 *10^-24 as an artifact of the numerical method ('numerical noise').
Posted 9 years ago
 Where do you get the z from?
Posted 9 years ago
 1722 seconds on my laptop, for a very messy result. I wonder if we're solving the same integral. Timing[ Integrate[(x^2 + y^2)*Boole[(x - z)^2 + (y - z)^2 <= 1], {x, -1, 3}, {y, -1, 3}, {z, 0, 2}] ] 
Posted 9 years ago
 took 11.5 seconds on my computer.
Posted 9 years ago
 Me too, but it's an interesting 3D problem.
Posted 9 years ago
 Integrate[(x^2 + y^2)*Boole[(x - z)^2 + (y - z)^2 <= 1], {x, -1, 3}, {y, -1, 3}, {z, 0, 2}]. I left my notebook computing about a half an hour and nothing happened.
Posted 9 years ago
 Wow! Very good. Thanks!
Posted 9 years ago
 In:= Integrate[ Sqrt[4 - x^2 - y^2]* Boole[x^2 + y^2 - 2 x <= 0], {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \[Infinity]}] Out= 8/9 (-4 + 3 \[Pi])