# Evaluating an integral over an elliptical disk

Using Mathematica, I am trying to evaluate the integral $$I=\iint_A xy\,\mathrm{d}x\,\mathrm{d}y$$ over the region $A=\{(x,y): ax^2+2hxy+by^2\le r^2\}$ where $a>0, ab-h^2>0$.

While trying to do this by hand, I changed variables $(x,y)\to(u,v)$ to get $$I=\iint_{u^2+v^2\le r^2}\left(\frac{uv}{\sqrt{ab-h^2}}-\frac{hv^2}{ab-h^2}\right)\frac{1}{\sqrt{ab-h^2}}\,\mathrm{d}u\,\mathrm{d}v$$

Now I used this code but it doesn't work:

f[u_, v_] = (u v)/Sqrt[a b - h^2]-(h v^2)/(ab-h^2);
Integrate[f[u, v]*Boole[ u^2 + v^2 <= r^2],
{u, -r, r}, {v, -r, r}, Assumptions -> {a > 0 , ab-h^2 > 0}]


By hand, I got $\displaystyle I=-\frac{\pi r^4h}{4(ab-h^2)^{3/2}}$ and I need to check my answer.

I am aware that several of these questions have been asked here before like this one, to which I reffered. I am a novice as a user of Mathematica and I would like to know how to evaluate the likes of these integrals. I changed variables to $(u,v)$ so that I can apply a polar transformation at the end while doing this by hand. But is there a way to directly evaluate general integrals like this in Mathematica without changing variables in the first place? If you guide me to a link where this has been answered before then that would be helpful too.

Here are some other links I found but I am not sure which code to use here:

I am using Mathematica 7.0.

You've inconsistently put in/left out spaces, a b vs. ab. Also you need the assumption r > 0.

Integrate[((u v)/Sqrt[a b - h^2] - (h v^2)/(a b - h^2)) Boole[u^2 + v^2 <= r^2],
{u, -r, r}, {v, -r, r},
Assumptions -> {a > 0, a b - h^2 > 0, r > 0}]
(*  (h π r^4)/(4 (-a b + h^2))  *)


Using regions, which gives the answer the OP derived by hand:

Integrate[
x y,
{x, y} ∈ ImplicitRegion[a x^2 + 2 h x y + b y^2 <= r^2, {x, y}],
Assumptions -> {a > 0, a b - h^2 > 0, r > 0}]
(*  -((h π r^4)/(4 (a b - h^2)^(3/2)))  *)


Here's a way that might work in V7, but I cannot check. I relied on the form of cylindrical decomposition that Reduce returned, which in this case (of an ellipse) has a simple logical structure. (There is a way to deal with more complicated cylindrical decompositions that I've used elsewhere, but this was simpler and more easily understood.)

dom = Sequence @@ Cases[
Reduce[
a > 0 && a b - h^2 > 0 && r > 0 && a x^2 + 2 h x y + b y^2 < r^2,
{x, y}, Reals],
_[a_, ___, v : x | y, ___, b_] :> {v, a, b}, (* inequality to iterator *)
Infinity]
(*
Sequence[
{x, -Sqrt[((b r^2)/(a b - h^2))], Sqrt[(b r^2)/(a b - h^2)]},
{y, -((h x)/b) - Sqrt[(b r^2 - a b x^2 + h^2 x^2)/b^2],
-((h x)/b) + Sqrt[(b r^2 - a b x^2 + h^2 x^2)/b^2]}
]
*)

Integrate[x y, dom, Assumptions -> {a > 0, a b - h^2 > 0, r > 0}]
(*  -((h π r^4)/(4 (a b - h^2)^(3/2)))  *)

• I am sorry to bother. Thank you. Apr 8 '18 at 11:57
• @StubbornAtom It's no bother. :) Apr 8 '18 at 11:58
• So is there no way to evaluate this without changing variables? The difficulty I guess is that I cannot determine the separate ranges of $x$ and $y$. Is that correct? Apr 8 '18 at 12:10
• Both give the same answer. So don't think there is anything wrong with the change of variables. Anyway, you have answered all my queries. Apr 8 '18 at 12:32
• @StubbornAtom OK, I see. A square-root constant is left off the first integral. Apr 8 '18 at 12:47

Another substitution-free method:

Simplify[Integrate[x y, {x, y} ∈ Ellipsoid[{0, 0}, Inverse[{{a, h}, {h, b}}/r^2]]],
a > 0 && b > 0 && r > 0 && a b > h^2]
-((h π r^4)/(4 (a b - h^2)^(3/2)))

• Is it possible that I may not get the desired output in an older version of Mathematica using your code? Apr 8 '18 at 12:42
• You didn't indicate what version you were using in your question. Please edit your question to include that information. Apr 8 '18 at 12:49
• I am using version 7.0. I will keep that in mind when I post next time. Apr 8 '18 at 12:59
• Ah, version 7... I don't think there's a substitution-free route there; Michael's route of rescaling the ellipse to a disk + converting to polar coordinates would be your best bet. Apr 8 '18 at 13:05
• Many thanks.... Apr 8 '18 at 13:06