Thread: Evaluate using spherical coordinates.

Evaluate using spherical coordinates:


I can represent this in cartesian coordinates like this:



and from here convert to spherical coordinates. I'm not sure how to completely convert this to spherical coordinates.

The only thing I could see was that on the xy-plane, part of the area is a quarter of a circle in the first quadrant with radius 1 (since y goes from 0 to and x goes from 0 to 1), so would go from 0 to .

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 ?

is the part of above the xy-plane. As an aid to seeing what that is, note that when y= 0, it gives or and similarly for y. That is, this is a cone making an angle 0f with the z-axis. The other boundary, is the part of or above the xy-plane. That is, of course, a sphere of radius . Yes, goes from 0 to . goes from 0 to and from 0 to .