28.6. EXERCISES 565

11. Evaluate∫

S (1+ x) dA where S is the part of the plane 4x+ y+ 3z = 12 which is inthe first octant.

12. Evaluate∫

S (1+ x) dA where S is the part of the cylinder x2 + y2 = 9 between z = 0and z = h.

13. Evaluate∫

S (1+ x) dA where S is the hemisphere x2 +y2 + z2 = 4 between x = 0 andx = 2.

14. For (θ ,α) ∈ [0,2π]× [0,2π] ,let

f (θ ,α)≡ (cosθ (4+ cosα) ,−sinθ (4+ cosα) ,sinα)T .

Find the area of f ([0,2π]× [0,2π]). Hint: Check whether fθ ·fα = 0. This mightmake the computations reasonable.

15. For (θ ,α) ∈ [0,2π]× [0,2π], let

f (θ ,α)≡ (cosθ (3+2cosα) ,−sinθ (3+2cosα) ,2sinα)T , h(x) = cosα,

where α is such that x= (cosθ (3+2cosα) ,−sinθ (3+2cosα) ,2sinα)T . Find∫f([0,2π]×[0,2π]) hdA. Hint: Check whether fθ ·fα = 0. This might make the compu-

tations reasonable.

16. For (θ ,α) ∈ [0,2π]× [0,2π], let

f (θ ,α)≡ (cosθ (4+3cosα) ,−sinθ (4+3cosα) ,3sinα)T , h(x) = cos2θ ,

where the parametrical description of the surface is

x= (cosθ (4+3cosα) ,−sinθ (4+3cosα) ,3sinα)T

Find∫f([0,2π]×[0,2π]) hdA. Hint: Check whether fθ ·fα = 0. This might make the

computations reasonable.

17. In spherical coordinates, φ = c,ρ ∈ [0,R] determines a cone. Find the area of thiscone.

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.