1
$\begingroup$

Evaluate $\int(x^2+y^2)^{1/2}dA$ where $D$ is region enclosed by the two circles: $x^2+y^2=64$ and $x^2+(y-4)^2=16$.

I'm confused on what the limits of integration for the corresponding double integral will be once converted to polar coordinates?

$\endgroup$
1
$\begingroup$

You're integrating the region between two circles, one with centre at the origin and radius 8, one with centre at $(0,4)$ and radius 4. To solve the problem, find the angles of the points where the circles intersect to get the limits of integration for $\theta$ and use the equations of the circles as your upper and lower limits of integration for the radius.

$\endgroup$
  • $\begingroup$ So for theta would it be 0 --> 2pi? $\endgroup$ – Jim JJ Apr 5 '16 at 0:25
  • $\begingroup$ No, the area you are integrating is somewhat like the intersection of a venn diagram visually. Find the points at which the two circles intersect, and then figure out the angles of these points using trigonometry. $\endgroup$ – Bob Roberts Apr 5 '16 at 0:29
  • $\begingroup$ Hint: when the points intersect, $x^2 + y^2 = 64 = 8y$ $\endgroup$ – Bob Roberts Apr 5 '16 at 0:30
  • $\begingroup$ Isn't the only point where the two circles intersect at (0,8)? $\endgroup$ – Jim JJ Apr 5 '16 at 0:43
  • $\begingroup$ My mistake, it is not a venn diagram. I presume then that the question is lacking for the area in the first circle, but not the second. In that case, there are a few ways to approach this, the easiest being to integrate over the area of the first circle, and then subtract the integral over the area of the second circle. $\endgroup$ – Bob Roberts Apr 5 '16 at 0:55

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.