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I want to calculate this double integral using Mathematica:

$$\iint_D x^3 y^3 \ln(1+x^2+y^2) \, \mathrm{d}x \, \mathrm{d}y,$$

where

$$D=\left\{ (x,y): x^2 + y^2 \leqslant 4, \; x,y \geqslant 0 \right\}.$$

I know how to calculate a definite double integral in Mathematica, but I am not sure how to do it with region $D$.

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1 Answer 1

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Try this:

Integrate[
 x^3*y^3*Log[1 + x^2 + y^2]*
  Boole[x^2 + y^2 <= 4 && x >= 0 && 
    y >= 0], {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \
\[Infinity]}]

(*  5/288 (-28 + 153 Log[5])  *)

Have fun!

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    $\begingroup$ Alternatively Integrate[ x^3*y^3*Log[1 + x^2 + y^2], {x, y} \[Element] Disk[{0, 0}, 2, {0, Pi/2}]] can also be used. $\endgroup$
    – SHuisman
    May 24, 2023 at 12:16

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