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

Posted 11 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 11 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
Posted 11 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]) 
Posted 11 months ago
 Yes with Assumptions work in my copy to.Thanks.
Posted 11 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.
Posted 11 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
Posted 11 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
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?
Posted 11 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*) 
 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....