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[✓] Calculate a double Integrate over an ellipse region?

Posted 11 months ago
10 Replies
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Hello! I have a simple question? How do I double integrate a function over an Elipsis Region? Something similar to a Circle Region, like

enter image description here

Thanks in advance.

10 Replies


 Integrate[1, {x, y} \[Element] Circle[{0, 0}, {a, b}]]

but Mathematica 11.3 gives unevaluated answer:

enter image description here

it should be: $4 a E\left(1-\frac{b^2}{a^2}\right)$

You can use:

ArcLength[Circle[{0, 0}, {a, b}]]
(* 4 b EllipticE[1 - a^2/b^2] *)


Works in my copy of 11.3

In[6]:= Assuming[a > 0 && b > 0, Integrate[1, {x, y} \[Element] Circle[{0, 0}, {a, b}]]]

Out[6]= 2 (b EllipticE[1 - a^2/b^2] + a EllipticE[1 - b^2/a^2])

Yes with Assumptions work in my copy to.


It also works without Assumptions, but returns (understandably) a much larger result and takes quite a bit of time. It might return unevaluated on a slower machine, I suppose.

Indeed it does:

enter image description here

Sorry for the late posting, but apparently every contribution to any thread only shows up much later for me than for many others on the Community.



Well, these might work, too:

Integrate[1, {x, y} \[Element] RegionBoundary[Ellipsoid[{0, 0}, {r1, r2}]], Assumptions -> {r1 > 0, r2 > 0}]

enter image description here

This gives the same answer (on 11.3)

Integrate[1, {x, y} \[Element] Circle[{0, 0}, {r1, r2}], Assumptions -> {r1 > 0, r2 > 0}]



Posted 11 months ago

Thanks for the response! But, I have another doubt, I was expecting a number as a result (like a volume). What Do I Have to do with EllipticE?

You will have to provide values for the two radii:

Integrate[1, {x, y} \[Element] Circle[{0, 0}, {1, 2}]]
(*8 EllipticE[3/4]*)


NIntegrate[1, {x, y} \[Element] Circle[{0, 0}, {1, 2}]]



PS: Note that this is the circumference not the "volume" as you suggest in your question. The area is easy:

Integrate[1, {x, y} \[Element] Disk[{0, 0}, {r1, r2}], Assumptions -> {r1 > 0, r2 > 0}]
(*\[Pi] r1 r2*)


NIntegrate[1, {x, y} \[Element] Disk[{0, 0}, {1, 2}]]

The volume of an Ellipsoid would be

Integrate[1, {x, y, z} \[Element] Ellipsoid[{0, 0, 0}, {r1, r2, r3}], Assumptions -> {r1 > 0, r2 > 0, r3 > 0}]
(*4/3 \[Pi] r1 r2 r3*)

or e.g.

NIntegrate[1, {x, y, z} \[Element] Ellipsoid[{0, 0, 0}, {1, 2, 3}]]
Posted 11 months ago

I don't know if I undestood correctly. How would you calculate this, for example?

enter image description here


this forum is not actually suppose to solve homework problems, but

Integrate[1, {x, y} \[Element] ImplicitRegion[x^2 + y^2 - 2 y <= 0, {x, y}]]

So, the answer in this case is Pi.



PS: Also, could you change the title of this thread to "Double Integral"? Doube has a different meaning....

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