30.2. EXERCISES 597

30.2 Exercises1. Find

∫S xdS where S is the surface which results from the intersection of the cone

z = 2−√

x2 + y2 with the cylinder x2 + y2 −2x = 0.

2. Now let n be the unit normal to the above surface which has positive z componentand let F (x,y,z) = (x,y,z). Find the flux integral

∫SF ·ndS.

3. Find∫

S zdS where S is the surface which results from the intersection of the hemi-sphere z =

√4− x2 − y2 with the cylinder x2 + y2 −2x = 0.

4. In the situation of the above problem, find the flux integral∫

SF ·ndS where n is theunit normal to the surface which has positive z component and F = (x,y,z).

5. Let x2/a2+y2/b2 = 1 be an ellipse. Show using Green’s theorem that its area is πab.

6. A spherical storage tank having radius a is filled with water which weights 62.5pounds per cubic foot. It is shown later that this implies that the pressure of thewater at depth z equals 62.5z. Find the total force acting on this storage tank.

7. Let n be the unit normal to the cone z =√

x2 + y2 which has negative z componentand let F = (x,0,z) be a vector field. Let S be the part of this cone which liesbetween the planes z = 1 and z = 2.

Find∫

SF ·ndS.

8. Let S be the surface z = 9− x2 − y2 for x2 + y2 ≤ 9. Let n be the unit normal to Swhich points up. Let F = (y,−x,z) and find

∫SF ·ndS.

9. Let S be the surface 3z = 9− x2 − y2 for x2 + y2 ≤ 9. Let n be the unit normal to Swhich points up. Let F = (y,−x,z) and find

∫SF ·ndS.