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# double integral on non-rectangular area

Posted 10 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 10 years ago
 Looking at my messy result, it is actually a sum of several integrals which Mathematica returned unevaluated.
Posted 10 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 10 years ago
 Where do you get the z from?
Posted 10 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 10 years ago
 took 11.5 seconds on my computer.
Posted 10 years ago
 Me too, but it's an interesting 3D problem.
Posted 10 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 10 years ago
 Wow! Very good. Thanks!
Posted 10 years ago
 In[1]:= Integrate[ Sqrt[4 - x^2 - y^2]* Boole[x^2 + y^2 - 2 x <= 0], {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \[Infinity]}] Out[1]= 8/9 (-4 + 3 \[Pi])