Question

How do you sketch `f(x,y)=arcsin(x^2+y^2-2)`?

Answer 1

Hello !

Let S be the surface of equation `z = \text{arcsin}(x^2+y^2-2)`.

S is a surface of revolution because `z = F(r)` where `r=sqrt{x^2+y^2}`. Here, `F(r) = \text{arcsin}(r^2-2)`.

First, you study the curve of equation `z = \text{arcsin}(x^2-2)` : you get

Second, you rotate this curve around (0z) axis and you get the surface S :

Remark that `f` exists only on the domain defined by `1\leq x^2+y^2\leq 3` : it's a disk of radius `sqrt{3}` with an circular hole of radius 1.