Using the Disk[] Primitive in Integration

Questions
1

Hi;

I am having difficulty using the Disk[] primitive to define my integration region - see attached. This method is quite new to me, however it seems really elegant, and I would sure like to learn how to use it correctly.

Thanks,
Mitch Sandlin

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2

All the examples I can find for using region-like primitives as integration regions use Cartesian coordinates rather than polar. So your integral would be written

In[95]:= Integrate[Cos[Sqrt[x^2 + y^2]] , {x, y} \[Element] Disk[{0, 0}, 1]]

Out[95]= 2 \[Pi] (-1 + Cos[1] + Sin[1])
3

I’m not sure how mathematically rigorous this is, but it seems to me that if you want to change the coordinate system, then you need to change the region:

Integrate[Cos[r] r, {r, t} \[Element] Rectangle[{0, 0}, {1, 2  Pi}]]
4

You can try IntegrateChangeVariables:

IntegrateChangeVariables[
 Inactive[Integrate][Cos[Sqrt[x^2 + y^2]],
  Element[{x, y}, Disk[]]],
 {r, \[Theta]}, "Cartesian" -> "Polar"]