drawing the area will help you how I draw it first I begin with the x-axis x changes from 0 to 1 as the drawing

then we will draw y , y changes from 0 to sqrt{1-x^2} . sub x=0 and x=1 which will give us quarter of circle in xy plane in the positve for xy

now z the hardest one first let see what is sqrt{x^2+y^2} = z if x=0 , y=1 , z=1 and if y=0,x=1,z=1

if z=1 this will give us a circle in xy plane with radius 1 but we interested just in xy positive

now we will see z= sqrt{2-x^2-y^2 } , if x=0,y=0, then z=sqrt{2}

it is sphere but positive z half of the sphere but since x,y are positive too so it is the quarter of the sphere

z=sqrt{2-x^2-y^2} will intersect with z=sqrt{x^2+y^2} at z=1 the circle in xy plane

the area is the area with the purple color

now can you determine the boundaries ?