Begin2.DVI

7-21. Find a unit normal vector to the cylinder x^2 +y^2 = 1

7-22. Find a unit normal vector to the sphere x^2 +y^2 +z^2 = 1

7-23. Find a unit normal vector to the plane ax +by +cz =d

7-24. Evaluate the surface integral

∫∫

R

x^2 yz d S
over the cylinder x^2 +y^2 = 1 lying in

the first octant between the planes z= 0 and z= 2.

7-25. Evaluate the surface integral

∫∫

R

xyz d S
where integration is over the upper

half of the sphere x^2 +y^2 +z^2 = 1 in the first octant.

7-26. Evaluate the surface integral

∫∫

R

F
·dS,
where F
=xˆe 1 +zˆe 2 and S is the

surface of the cylinder x^2 +y^2 = 1 between the planes z= 0 and z= 2.

7-27. Evaluate the surface integral

∫∫

R

F
·dS,
where F
=zˆe 1 +zˆe 2 +xy ˆe 3 and Sis

the upper half of the sphere x^2 +y^2 +z^2 = 1 lying in the first octant.

7-28. Evaluate the surface integral

∫∫

R

F
·dS,
where F
=eˆ 3 and Sis the upper half

of the sphere x^2 +y^2 +z^2 = 1.

7-29. Evaluate the surface integral

∫∫

R

F
×dS,
where F
= (z+y)ˆe 1 +x^2 ˆe 2 −yˆe 3

and Sis the surface of the plane z= 1,where 0 ≤x≤ 1 and 0 ≤y≤ 1.

7-30. Show that any curve on a surface defined by the parametric equations

x=x(u, v ) y=y(u, v ) z=z(u, v )

has an element of arc length given by

ds^2 =E du^2 + 2 F du dv +G dv^2 ,

where E, F, G are defined by the equations (7.54).

7-31.

(a) Show that when the curve z=f(x), x 0 < x < x 1 ,is rotated 360 ◦ about the z-axis,

the surface formed has a surface area

S=

∫ 2 π

0

∫x 1

x 0

r

√

1 + [f′(r)]^2 dr dθ

(b) The curve z=f(x) = HRxfor 0 ≤x≤Ris rotated 360 ◦about the zaxis. Find the

surface area generated.