Quick question on Polar Coordinates.
When evaluating the double integral and changing variables, I'm not sure if the limits are correct.
The question is as follows:
Evaluate $$\int\!\!\!\int_D xy\sqrt{x^2 + y^2}\,dxdy $$where $D = \{(x,y) \mid 1 \leq x^2 + y^2 \leq 4,\ x \geq 0,\ y \geq 0\}$
So my question is when I change to polar coordinates, is the limit for the integral with respect to r from 1 to 2 or 1 to 4?
Instinctively, I would say it's 1 to 4 but the answer given out by the lecturer (which does not have all the steps) has the limts at 1 to 2.
Is it maybe because $x^2 + y^2 = a^2$?
Note: I have the new integral, in terms of r and $\theta$ as:
$\int$$\int$$r^4$cos$\theta$sin$\theta$drd$\theta$