Math 22 Review (15.3-16.3)
1. Let R be the planar region bounded by
, 2 , & 2y x y x x
. Evaluate
3
R
xdA
2. Let R be the planar region outside of
2 2 1x y
and inside of
2 2
( 1) 1x y
. Find the
volume of the solid above R and below the graph of
2 2
( , )f x y x y
.
3. Evaluate: (a)
2
4 2
0 /2
x
y
e dxdy
(b)
2
2 0 2 2 3/ 2
0 4
( )
x
x y dydx
4. Let
R
be the planar region bounded by
2y
,
siny x
,
0x
and
x
. Suppose that the
density at any point on
R
equals the distance from that point to the y-axis.
a. Find the mass of
R
.
b. Find the center of mass.
c. SET up the integral that could be used to find the moment of inertia of
R
about the y-
axis.
5. Let
R
be the solid region bounded by
0, 0, 1, & 2 2 5x y z x y z
. Evaluate
RxdV
.
6. Let
R
be the solid region bounded below
2 2 2 4x y z
and above
2 2
3z x y
. SET UP an
iterated integral (or integrals) for
R
yzdV
in
(a) cylindrical coordinates; (b) spherical coordinates.
7. R is the region bounded by the ellipse
2 2
9 4 36x y
. Let
2x u
and
3y v
.
a. Sketch the corresponding region in the uv-plane for this substitution.
b. Use the substitution to rewrite the integral
2
Rx dA
.
8. Sketch the vector field
( , ) ( ) ( )F x y x y x y i j
.
9. Suppose that the base of a fence is the portion of the cardioid
1 sinr
in the first quadrant
and the height of .the fence at
( , )r
is
. SET UP an integral that could be used to find the
area of the fence.
10. Let C be the graph of
( ) cos( ) sin( ) , , 0 1 / 2t t t t t r i j k
. Evaluate the line integral
Cxzds
.
11. Determine if the following vector fields are conservative.
a.
( , ) (2 3 ) ( 3 4 8)x y x y x y F i j
b.
( , ) ( cos ) ( sin )
x x
x y e y e y F i j
12. Let C be the straight path from
(1,1,0)
to
( 1,2,1)
. Find the work done by
3 2 2
( , ) (ln 2 ) (3 / )x y y xy x y x y F i j
along C.
13. Show that the line integral
(1 )
x x
C
ye dx e dy
is independent of path.