# Disagreement between Integration over Region and Contour integral

I am trying to do a line integral along an ellipse. The am integrating the field $$x^2 + y^2$$. I want to integrate over an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where the perimeter of the ellipse is $$2\pi$$ and $$a = 1.5$$. To get $$b$$ such that the perimeter is $$2\pi$$, I use Ramanujans equation. Here is the code I use to do this integral over a region:

a = 1.5;
b =  1/3 (3 - 2 a + Sqrt[3 + 6 a - 5 a^2]);
R = ImplicitRegion[x^2/a^2 + y^2/b^2 == 1., {x, y}];
NIntegrate[x^2 + y^2, {x, y} \[Element] R]


This integral returns 5.397.

I can also parameterize the ellipse using $$x = a\cos{\theta}$$ and $$y = b\cos{\theta}$$. Then I would plug this into the field and integrate over $$\theta$$:

a = 1.5;
b =  1/3 (3 - 2 a + Sqrt[3 + 6 a - 5 a^2]);
NIntegrate[(a*Cos[\[Theta]])^2 + (b*Sin[\[Theta]])^2,{\[Theta],0,2*Pi}]


This integral gives me 7.33. Why are the two integrals returning different results?

• By the way, in this case you could integrate directly over the (boundary of) ellipse: Integrate[x^2 + y^2, {x, y} \[Element] Circle[{0, 0}, {a, b}]], and with exact a and b, get an exact answer. Nov 2, 2022 at 6:01

We need to add Jacobian in curve integral. Sqrt[D[a*Cos[θ], θ]^2 +D[b*Sin[θ], θ]^2] when we change the variables in integral.
a = 1.5;

5.39697