Find the surface area of x^2+y^2+z^2=9 that lies above the cone z= sqrt(x^@+y^2)

The cone equation gives



which means that the intersection of the cone and sphere occurs at



i.e. along the vertical cylinder of radius when .

We can parameterize the spherical cap in spherical coordinates by



where and , which follows from the fact that the radius of the sphere is 3 and the height at which the sphere and cone intersect is . So the angle between the vertical line through the origin and any line through the origin normal to the sphere along the cone's surface is



Now the surface area of the cap is given by the surface integral,