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Quiz #7
NAME:
1 2
#1. (5 pts) Evaluate the integral
  2xy dy dx
0 0
#2(6 pts) Rewrite the integral as an integral or sum of integrals with the order of operation reversed.
It is probably a good idea to sketch the region that is determined by the limits of integration.

2
sin( x )
 2
1
 
f ( x, y ) dy dx
#3(6 pts) Use a change of variables to convert the integral
1

1 x 2
0  1 x
2

2 y
0
xy dz dy dx into cylindrical
coordinates. Do not evaluate the integral.
#4(8 pts) Evaluate
 ( x
D
2
 y 2 )dA , where D is the region in the first quadrant bounded by the
graphs of y  x  0, y  x  1, xy  1, and xy  2 .
#5. Consider the integral
2

0
0
1 
x2
4


2
3 1 y 2  x4
0
 f ( x, y, z ) dz dy dx . Rewrite this as an equivalent
iterated integral by sketching the region of integration, then putting appropriate limits in the integral
below:
a. (3 pts) Sketch the region:
b. (4 pts) Put in the correct limits of integration for this integral with the order of integration as
written. Write legibly – you may find it easier to re-write the entire integral if I haven’t left
enough room for the limits.
2

0
0
1 
x2
4


2
3 1 y 2  x4
0
 f ( x, y, z ) dz dy dx 



f ( x, y, z ) dx dy dz
 xu
#6(8 pts) Let T be the transformation given by: 
 y  v(1  u )
.
a.
If D* is the region [0, 2][0, 2] in the uv-plane, determine D, the image of D* in the
xy-plane.
b.
Calculate

D
y
1

e x
dA by changing variables to uv-coordinates.
1 1 x
#7(2 pts) Evaluate

0
0
 yx 


e y  x  dy dx