Calculate the area of the surface Y given by the equation $z = x^2 + y^2 − 1$ when $z ≤ 0$
Here is my solution:
I got the answer correct but there is something I just did (not toally randomly but I ignored certain requirements) which was the boundaries. In the question it says that $z\leq0$ but my boundaries where $0 \leq r \leq 1$ and $0 \leq \theta \leq 2\pi$ and I was wondering how in the hell my solution took into account that z is supposed to be less than 0?
You implicitly chose the boundaries of your parameterized integral in a way that satisfied the original condition. Indeed,
$$z\leq0\iff x^2 + y^2 \leq 1\iff r\leq 1.$$