$$\oint \left(\frac {\arctan(x^3+3)}{3+x^2} +2\cos(x^2+y^2)\right)\,dx +\left(\frac{\sin(\cos^2(1+y^2)}{y^2+100}+2y\cos(x^2+y^2)\,dy\right)$$

where $C$ is positively oriented and is the ellipse $(x-4)^2 + y^2/4=1$


After finding the partials, I get the double integral $$\iint -8xy\sin(x^2+y^2)\,dA$$

After this I'm not sure what limits to use.

  • $\begingroup$ Can you show me your differentiation? I am getting $\iint 4y(x-1)\sin(x^2 + y^2) \ dA$ $\endgroup$
    – Doug M
    Commented Apr 19, 2018 at 16:22

1 Answer 1


The double integral is over the "elliptic disk" that has that ellipse as boundary. This is, of course, the ellipse with center at (4, 0) and "semi-axes" of length 1, in the x direction, and 2, in the y direction. Further, solving the equation of the ellipse for y, $y= \pm2\sqrt{1- (x- 4)^2}$. So to cover the "elliptic disk" x must go from 4- 1= 3 to 4+ 1= 5 and, for each x, y goes from $-2\sqrt{1- (x- 4)^2}$ to $2\sqrt{1- (x- 4)^2}$.


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