398 CHAPTER 21. THE INTEGRAL ON TWO DIMENSIONAL SURFACES IN R3

18. Let F = (x,y,z) and let S be the curved surface which comes from the intersectionof the plane z = x with the paraboloid z = x2 + y2. Find an iterated integral for theflux integral

∫SF ·ndS where n is the field of unit normals which has negative z

component.

19. Let F = (x,0,0) and let S denote the surface which consists of the part of the spherex2+y2+z2 = 9 which lies between the planes z= 1 and z= 2. Find

∫SF ·ndS where

n is the unit normal to this surface which has positive z component.

20. In the situation of the above problem change the vector field to F = (0,0,z) and dothe same problem.

21. Show that for a sphere of radius a parameterized with spherical coordinates so that

x = asinφ cosθ , y = asinφ sinθ , z = acosφ

the increment of surface area is a2 sinφdθdφ . Use to show that the area of a sphereof radius a is 4πa2.