The answers by Kitonum and Carl are carried out in Cartesian and Spherical coordinates, respectively. For the sake of completeness, I mention here the integration in Cylinderical coordinates:

restart;

y := rho*sin(theta);
                             y := rho sin(theta)

int(y*rho, z = -sqrt(-rho^2+1) .. sqrt(-rho^2+1), rho = 0 .. 1, theta = 0 .. Pi);
                                    1   
                                    - Pi
                                    4   

1-Its not clear that what is the funtion that you want to integrate (Integrand)!

2-Also, the domain of integration that you want to integrate over it! (Limits of Integration)

3-The usage of arguments of the "int" command is also wrong.

I think that you mean to integrate the funtion "sqrt(1-x^2-z^2)" over the domain "x^2+z^2<=1". In this case, you should do the following:


restart;
f := sqrt(-x^2-z^2+1);

                                                 (1/2)
                                 /  2    2     \     
                          f := \-x  - z  + 1/   
 
int(int(f, z = -sqrt(-x^2+1) .. sqrt(-x^2+1)), x = -1 .. 1);
                                    2   
                                    - Pi
                                    3

As you see, I found the answer by hand. I am wondering that why maple cannot compute this integral. It's not so complicated! :/