0
$\begingroup$

Let γ be the ellipse $x^2 + 4y^ 2 = 4$, oriented anticlockwise. Compute $\int_c(4y − 3x)dx + (x − 4y)dy$

I used green theorem with P and Q. and got $$-3\int\int_\ dxdy$$

The answer is $-6\pi,$ so how do I get $2\pi$ from the double integral? Or am I just tackling the problem all wrong?

$\endgroup$
  • $\begingroup$ green is Green: always use a Capital for names. $\endgroup$ – Jean Marie Mar 22 '17 at 22:36
1
$\begingroup$

The double integral will give you the area of the ellipse.

Your ellipse is $$\left( \frac{x}{2}\right)^2+\left(\frac{y}{1}\right)^2 =1 $$ with $a=2$ and $b=1$ and has area $\pi a b= 2 \pi$

$\endgroup$
  • $\begingroup$ what would the bounds be though? isn't 2pi usually expressed in term of d(theta) not dx or dy? $\endgroup$ – user3427042 Mar 22 '17 at 22:04
  • $\begingroup$ dx dy is a 2d cartesian area element, and the double integral is over the entire ellipse so it is calculating the area of the ellipse. You don't need to change to another coordinate system. $\endgroup$ – PM. Mar 22 '17 at 22:09
  • $\begingroup$ denominator should be 4 right not 2? $\endgroup$ – user3427042 Mar 22 '17 at 22:11
  • $\begingroup$ A pleasure. You can see the details of the double integral, in both cartesian and polar coordinates, here mymathforum.com/calculus/29970-area-ellipse.html $\endgroup$ – PM. Mar 22 '17 at 22:16
  • $\begingroup$ Regarding denominator, 2^2 = 4. $\endgroup$ – PM. Mar 22 '17 at 22:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.