SECTION 2.7 EXERCISES

Review Questions

23-26. Solve and compute Jacobians Solve the following relations for

1. Suppose S is the unit square in the first quadrant of the uv-plane.

x and y, and compute the Jacobian J(u, v).

Describe the image of the transformation T: x = 2u, y = 2v.

23. u = x + y, v = 2x - y

24. u = xy, v = x

2.

Explain how to compute the Jacobian of the transformation

T: x = 8(u, v), y = h(u, v).

25. u = 2x - 3y, v =y-x

26. u = x + 4y, v = 3x + 2y

3.

Using the transformation T: x = u + v, y = u - v, the image of

27-30. Double integrals-transformation given To evaluate the fol-

the unit square S = { (u, v): 0 S u = 1, 0 = v = 1} is a region

lowing integrals, carry out these steps.

in the xy-plane. Explain how to change variables in the integral

a. Sketch the original region of integration R in the xy-plane and the

JRf(x, y) dA to find a new integral over S.

new region S in the uv-plane using the given change of variables.

b. Find the limits of integration for the new integral with respect to u

4.

Suppose S is the unit cube in the first octant of uvw-space with

and v.

one vertex at the origin. What is the image of the transformation

c. Compute the Jacobian.

T: x = u/2, y = v/2, z = w/2?

d. Change variables and evaluate the new integral.

Basic Skills

5-12. Transforming a square Let S = { (u, v): 0 S us 1,

27 .

xy dA, where R is the square with vertices (0, 0), (1, 1),

O S v s 1} be a unit square in the uv-plane. Find the image of S in

(2, 0), and (1, -1); use x = u + v, y = u - v.

the xy-plane under the following transformations.

5. T: x = 2u, y = v/2

28 .

// x 3y dA , where R = { ( x , y ) : 0 5 x 5 2, x sysx+ 4);

6. T: x = -u,y = -v

use x = 2u, y = 4v + 2u.

7. T: x = (u + v)/2,y = (u-v)/2

8. T: x = 2u + v,y = 2u

29 .

x 2 Vx + 2y dA, where

9. 1:x = U2 - v2, y = 2uv

R

R = { ( x, y ) : 0 = x = 2, - x/ 2 = y = 1 - x}; use

10. T: x = 2uv, y = u2 - v2

x = 2u, y = v - u.

11. T: x = u Cos TV, y = u sin TV

30 .

xy da, where R is bounded by the ellipse 9x2 + 4y? = 36;

12. T: x = v sin Tru, y = V COS Tru

use x = 2u, y = 3v.

13-16. Images of regions Find the image R in the xy-plane of the re-

gion S using the given transformation T. Sketch both R and S.

31-36. Double integrals-your choice of transformation Evaluate

the following integrals using a change of variables. Sketch the original

13. S = { ( u, v) : v = 1 - u, u 2 0, v 2 0 }; T: x = u,y = v2

and new regions of integration, R and S.

14. S = { (u, v): u2 + v2 = 1}; 1:x= 2u,y=4v

15. S = { ( u, v) : 1 = u < 3, 2 = v = 4}; 1: x = u/v,y = v

31. J J Vx - vaxdy

16. S = { ( u, v ) : 2 = u < 3, 3 = v = 6}; 1: x = u,y = v/u

32 .

([ Vy2 - x2 dA, where R is the diamond bounded by

onily) .op

17-22. Computing Jacobians Compute the Jacobian J(u, v) for the

following transformations.

y - x = 0, y - x = 2, y+ x= 0, and y + x =2

17. T: x = 3u, y = -3v

33.

y - * dA, where R is the parallelogram bounded

18. T: x = 4v, y = -2u

// ( + 2x + 1)

R

19. T: x = 2uv, y = u2 - v2

by y - x = 1, y - x = 2, y + 2x = 0, and y + 2x = 4

20. T: x = u COS TTV, y = u sin TV

34.

ey dA, where R is the region bounded by the hyperbolas

21. T: x = (u+ v)/V2,y = (u-v)/V2

xy = 1 and xy = 4, and the lines y/ x = 1 and y/ x = 3

22. T: x = u/v,y = v