It's been a while since I've done calc 3 work, and I'm a bit stuck on how to convert the integral $\iint f_{xyz}(x,y,z)\,dy\,dz$ into polar form.
I would like to use $y = r \cos \theta$ and $z = r \sin \theta$. From this, I tried to derive $dy$ and $dz$, but I'm pretty sure I'm not doing this correctly.
$$ \frac{dy}{dr} = \cos \theta \\ \therefore dy = \cos \theta\,dr \\ \frac{dz}{dr} = \sin \theta \\ \therefore dz = \sin \theta\,dr $$
If I plug this into my integral I get
$$ \iint f_{xyz}(x, r\cos \theta, r \sin \theta) \cos \theta \sin \theta \,dr\,dr $$
Which doesn't look right at all. I'm pretty sure the polar form should be $$ \iint f_{xyz}(x, r\cos \theta, r \sin \theta)\,r\,dr\,d\theta $$
Would the correct approach be to do something like $$ \frac{dz}{d\theta} = r \cos \theta \\ \therefore dz = r \cos \theta\,d\theta $$
Then the integral looks like
$$ \iint f_{xyz}(x, r\cos \theta, r \sin \theta) \cos^2 \theta\,r\,dr\,d\theta $$
But this doesn't look right either because of the $\cos^2 \theta$. Where am I going wrong with my thought process?
Also, I'm aware that we have $$ dA = dy\,dz = r\,dr\,d\theta $$
But here, I'm interested in understanding where I'm going wrong with my approach.