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

Posted 7 months ago
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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 Thanks in advance.
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Posted 7 months ago
 Hello  Integrate[1, {x, y} \[Element] Circle[{0, 0}, {a, b}]] but Mathematica 11.3 gives unevaluated answer: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] *) Regards,MI
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Posted 7 months ago
 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]) 
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Posted 7 months ago
 Yes with Assumptions work in my copy to.Thanks.
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Posted 7 months ago
 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.
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Posted 7 months ago
 Indeed it does: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. Cheers,Marco
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Posted 7 months ago
 Well, these might work, too: Integrate[1, {x, y} \[Element] RegionBoundary[Ellipsoid[{0, 0}, {r1, r2}]], Assumptions -> {r1 > 0, r2 > 0}] This gives the same answer (on 11.3) Integrate[1, {x, y} \[Element] Circle[{0, 0}, {r1, r2}], Assumptions -> {r1 > 0, r2 > 0}] Cheers,Marco
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Posted 7 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?
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Posted 7 months ago
 You will have to provide values for the two radii: Integrate[1, {x, y} \[Element] Circle[{0, 0}, {1, 2}]] (*8 EllipticE[3/4]*) and NIntegrate[1, {x, y} \[Element] Circle[{0, 0}, {1, 2}]] (*9.68845*) Cheers,MarcoPS: 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*) and NIntegrate[1, {x, y} \[Element] Disk[{0, 0}, {1, 2}]] (*6.28319*) 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}]] (*25.1327*) 
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Posted 7 months ago
 I don't know if I undestood correctly. How would you calculate this, for example?
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Posted 7 months ago
 Hi,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}]] (*\[Pi]*) So, the answer in this case is Pi.Cheers,MarcoPS: Also, could you change the title of this thread to "Double Integral"? Doube has a different meaning....
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