# Using the Disk\[\] Primitive in Integration

**URL:** <https://community.wolfram.com/t/using-the-disk-primitive-in-integration/23825>\
**Category:** Questions\
**Tags:** calculus\
**Created:** [April 2, 2024, 5:53pm UTC](https://community.wolfram.com/t/using-the-disk-primitive-in-integration/23825 "2024-04-02T17:53:18Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![masandlin](https://avatars.discourse-cdn.com/v4/letter/m/839c29/32.png) [@masandlin](https://community.wolfram.com/u/masandlin)\
**Post date:** [April 2, 2024, 5:53pm UTC](https://community.wolfram.com/t/using-the-disk-primitive-in-integration/23825/1 "2024-04-02T17:53:18Z")

</div>

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

[Open in Wolfram Cloud ↗](https://www.wolframcloud.com/obj/3d8b30b7-0689-4832-8812-8fe0886760a0)

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<div class="post-metadata">

**Author:** ![jasonb](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/jasonb/32/20741_2.png) [@jasonb](https://community.wolfram.com/u/jasonb)\
**Post date:** [April 2, 2024, 7:28pm UTC](https://community.wolfram.com/t/using-the-disk-primitive-in-integration/23825/2 "2024-04-02T19:28:12Z")

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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

```auto
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])

```

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<div class="post-metadata">

**Author:** ![eric3](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/eric3/32/12760_2.png) [@eric3](https://community.wolfram.com/u/eric3)\
**Post date:** [April 2, 2024, 7:34pm UTC](https://community.wolfram.com/t/using-the-disk-primitive-in-integration/23825/3 "2024-04-02T19:34:59Z")

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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}]]

```

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<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [April 3, 2024, 1:43pm UTC](https://community.wolfram.com/t/using-the-disk-primitive-in-integration/23825/4 "2024-04-03T13:43:47Z")

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You can try `IntegrateChangeVariables`:

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

```