21.6. EXERCISES 397

4. Find a parametrization for the straight line joining (1,3,1) and (−2,5,3).

5. Find a parametrization for the intersection of the surfaces 4y + 3z = 3x2 + 2 and3y+2z =−x+3.

6. Find the area of S if S is the part of the circular cylinder x2 + y2 = 4 which liesbetween z = 0 and z = 2+ y.

7. Find the area of S if S is the part of the cone x2 +y2 = 16z2 between z = 0 and z = h.

8. Parametrizing the cylinder x2+y2 = a2 by x = acosv,y = asinv,z = u, show that thearea element is dA = adudv

9. Find the area enclosed by the limacon r = 2+ cosθ .

10. Find the surface area of the paraboloid z = h(1− x2− y2

)between z = 0 and z = h.

Take a limit of this area as h decreases to 0.

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 θ is such that 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 compu-

tations reasonable.

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