I get the problem and can set it up but am struggling with the integration. Is there u-substitution and I just can't figure it out??
THE PROBLEM
A thin metal plate occupies a region D, which lies in the first quadrant and is bounded by the circles of radius $r=1$cm and $r=2$cm and the line $y=x$ and the y-axis
The density of the plate is given by the function $\delta(x,y)=y/\sqrt{x^{2}+y{2}}$ g/$cm^{2}$
Find the mass.
The shape of the region and the $\sqrt{x^2+y^2}$ in the definition of the density function both suggest that you should use polar coordinates, in which case
$$\delta(r,\theta)=\frac{r\sin\theta}{r^2}=\frac{\sin\theta}r\;,$$
since you’ll be dealing only with positive radii. You’re probably fine from here, but if not, I’ve left a little more in the spoiler-protected block.
Now your element of area is $r\,dr\,d\theta$, so you want $\int_{\pi/4}^{\pi/2}\int_1^2\sin\theta\,dr\,d\theta$, which is nice and straightforward.
