Change of Variables II
Reading Assignment: Read 5.5 again. Twice.
Suggested problems: 5.5: 18, 19, 20,21, 23, 27,28,29, 31, 32 (hard!);
Also: 5.8: 1, 3, 5, 9, 11, 12, 15, 17, 20
1) Find a transformation that will turn the triangle with vertices (0,0), (1,0), and (0,1) in the uvplane into the triangle with vertices (1,0), (0,1) and (2,2) in the xy-plane.
2) Find a transformation that will turn the unit square [0,1]  [0,1] in the uv-plane into the
triangle 0  y  x  1 in the xy-plane.
3) Calculate
1

x
0 0
xy dy dx both directly and by changing variables as suggested in problem 2).
4) Calculate
5) Convert
integrate.
2
4 x2
0
 4 x2

1  /4
2
0 0
0
 
sin( x 2  y 2 ) dy dx .
 3 sin(2 ) d d d  into an integral in rectangular coordinates. Do not
6) Calculate
1 1 x

0 0
e y /( x y ) dy dx by making a helpful change of variables. (Ignore any difficulty
this integral may have at the origin.) Hint: it would be nice to simplify the exponent, so why not
take it all as one chunk? The x + y alone is causing troubles, so maybe it ought to be chunked
together too.
7) Let R be the region in the first quadrant bounded by the hyperbolas xy = 1, xy = 3, x2 – y2 = 1,
and x2 – y2 = 4. By choosing a convenient change of variables, calculate  ( x2  y 2 ) dA
R
8) Evaluate each integral.
a.
b.

9- x2
2
4 x2
3
0
0

0
0

2
0

x 2  y 2 dz dy dx
4 x2  y 2
0
z 4  x2  y 2 dz dy dx
9) Let D be the interior of the parallelogram with vertices at (0, 0), (2, 1), (3, 2), and (1,3) . Use
dy dx
a change of variables to simplify your task of evaluating 
.
1 x  2 y
D
(1,3)
(3,2)
D
(0,0)
(2,-1)