Math 251 Review problems for chapter 13
J. Lewis
I] Evaluate each integral.
 ( x
a)
2
2
 y ) dA
R is the triangle with vert ices (0,0), (2,4) and (0,6).
R
 xe
b)
y
2
2
R is bounded by y  32  x and y  x  18
dA
R

c)
2
R is bounded by x  y and x  y  6
x dA
R
II] Evaluate each iterated integral in two orders when possible without a computer.
11

a)
 4
1
0 x1  y
2
dy dx
b)
1
 sec

8
2
x dydx
c)
3
 
0 1
0 tan x
x
e
y
2
dydx
x
4
2 x
d)
3 3
2

2
 sec( x ) tan( x ) dydx
e)
0 0

2
2
 sec ( y ) dydx
0 x
f)
e 1
1


0
ln( y  1)
1
y 1
dxdy
III] Evaluate each by converting to polar coordinates.
a)
x y

x
R
b)
dA where R is the region bounded by the polar graph of r  cos   sin  .
 y
2
y

x
R
c)
2
2
dA where R is the region bounded by the polar graph of r  cos   1 .
 y
6
6x x


0
0
2
2
1
2
x  y
dydx
2
IV] Find the mass and center of mass of the lamina described by the region R with the given density
function,  ( x , y ) .
a) R is the triangle with vertices (0,0), (0,1) and (1,1).  ( x , y )  e x .
b) R is bounded by y = x, y = 0, x = 1  ( x , y )  xe
y
V] Sketch the solid and find the volume of the region bounded by:
a) x
2
2
 y
 9,
y
c) z  x 2  y 2 ,
 z
( x  2)
z
d) Set up only:
2
2
 x
2
2
2
9
 y
2
2
 y ,
b) y  4  x 2 ,
 4,
2
st
z  4  x in the 1 octant
z  0
( x  1)
2
 ( y  1)
2
 2,
z  0
VI] Graph each polar equation in the x-y-plane.
a) r  cos 5 Find the area of one loop. Graph one loop only.
b) r  3 cos   4 sin  . Convert to Cartesian coordinates and identify it.
c) r  6 sin 
d) r  6 cos 
VII] Convert each equation to spherical coordinates and sketch.
a) z  A
c)
2
2
2
 B B>0
x  y , A > 0.
2
x  y
B) z  c , c > 0
VIII] Evaluate:
a)
 ( x 
y ) dV
2
where E is bounded by z  x ,
y  x,
y 3
x  0
E
b)
1

where E is bounded by y  0 ,
dV
2
y 9x ,
z 9x
y
E
IX] Evaluate using spherical co-ordinates:
a)

x
2
 y
2
2
 z dV where E is bounded by z 
E
2
b)
2 x


0
0
2
2
4 x  y
2
 z dzdydx
2
x y
2
3x
2
 3y
2
and x
2
 y
2
 z
2
1.
X] Evaluate using either cylindrical or spherical coordinates to find the volume of:
a) the region inside x
2
b) the region inside x
2
 y
2
 y
2
 z
2
 z
2
2
2
 4 z and inside 2 z  x  y .
 4 z and inside 3 z
2
2
2
 x  y .