MATH 223
Change Of Variables
ZZ
Problem 1. Use a change of variables to evaluate
(2y+x) dA, where
R
3
3
1
R is the region bounded by y = x + 2, y = x + 4, y = − x + 1 and
2
2
4
1
y = − x + 4.
4
Problem 2. Use a change of variables to find the area of the parallelogram P bounded by 2y = Z1Z− x, 2y = 3 − x, y = 3x, and y = 3x + 4.
Recall. The area of P is
dA.
P
ZZ
Problem 3. Use a change of variables to evaluate
cos(y + x) dA,
R
where R is the region bounded by y = 1 − 2x, y = 5 − 2x, y = 3x − 2
and y = 3x + 3.
ZZ
ey−3x
dA, where P is the region bounded
Problem 4. Evaluate
P x + 2y
by 2y = 1 − x, 2y = 3 − x, y = 3x, and y = 3x + 4.
Problem 5. Use a change of variables to find the volume of the solid
region bounded by the paraboloid z = x2 + y 2 and the elliptic cylinder
x2 y 2
+
= 1.
9
4
Problem 6. Find the area of the region R bounded by the curves xy = 1,
xy = 3, xy 1.4 = 1 and xy 1.4 = 2.
Problem 7. Find the area of the region R bounded by the curves y = x2 ,
y = 2x2 , x = y 2 and x = 4y 2 .
ZZ
1
Problem 8. Use a change of variables to evaluate
dA,
2
2 2
R (x + y )
where R is the region bounded by the circles x2 + y 2 = 2x, x2 + y 2 = 6x
and the circles x2 + y 2 = 2y and x2 + y 2 = 8y.