Need help converting $z = \ln(x^2 + y^2)$ to polar

Question

The full question is this:

Volume of a solid in any region R is given by: $$\int\!\!\!\int_Rf(x,y)dydx $$

where, $$f(x,y) = z = \ln(x^2+y^2)$$

and, $$x^2+y^2=r^2$$

There for, $$dydx = \_\_\_drd\theta$$

My answer is:
$$r\ln(r^2)\, dr\,d\theta$$

but the book says the correct answer is: $$r \,dr\,d\theta$$

How is that so?

Answer 1

There two different things: $\ln(x^2+y^2)$ is transformed into $\ln(r^2)$, and $dx\,dy$ is transformed into $r\,dr\,d\theta$.

Of course I cannot know, what exactly states yor book, but at least partially it tells the truth.

Answer 2

Oh my god, I was dumb.

I real realized the question was asking what $dxdy = r drd\theta$

It didn't ask me to convert $ln(x^2+y^2)dxdy to drd\theta$