1
Math E-21a – Fall 2009 – HW #13 problems
To be turned in Thurs, Dec 10
:
Section 12.6
:
2. Find the area of the part of the plane
25
1
0
xy
z
that lies inside the cylinder
22
9
.
4. Find the area of the part of the plane with vector equation
(,) 1 ,
2,3 5
uv
vu
v
u v
r
that is given by
01
,
.
6. Find the area of the part of the surface
2
13 2
zx
y
that lies above the triangle with vertices (0,0), (0,1),
and (2,1).
10. Find the area of the part of the paraboloid
x
yz
that lies inside the cylinder
9
.
[
Hint
: Try modifying the method for a graph
(, )
zf
x
y
for this case where we have
(,)
x fyz
.]
24. The figure shows the surface created when the
cylinder
1
intersects the cylinder
1
xz
. Find the area of this surface.
[
Hint
: Find the surface area of one of the eight
identical pieces of this surface, e.g. the part above
the triangular region bounded by the lines
y
x
,
y
x
, and
1
x
.]
28. The figure shows the torus obtained by rotating
about the
z
-axis the circle in the
xz
-plane with
center ( ,0,0)
b
and radius
ab
.
Parametric equations for the torus are
cos
cos
cos
sin
cos
sin
sin
xb
a
yb
a
za
where
and
are the angles shown in the figure.
Find the surface area of the torus.
Section 13.6
:
8. Evaluate the surface integral
S
xydS
where
S
is the triangular region with vertices
(1,0,0)
,
(0,2,0)
,
and
(0,0,2)
.
[
Note
: The surface
S
is a portion of a plane.]
12. Evaluate the surface integral
S
zdS
where
S
is the surface
2
2,0
1
,0
1
x
y
z