Problems 15,28, and 32!
27 Change of Variables in Multiple Integrals 189 As another example, suppose the region is bounded by the lines y = x (or y/x = 1) and y - 2x (or y/x = 2) and by the hyperbolas xy = 1 and xy = 3, Then the new vari- ables should be u - xy and v - y/x (or vice versa). The new region of integration is the rectangle S - { (u, v): I S u = 3, 1 = v s 2). 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 Jacoblan J(u, v). Describe the image of the transformation 7:x - 20. y - 2v. 23. 1 = x + y. V. 2 - y 24. u = xy, v = x 2. Explain how to compute the Jacobian of the transformation 25. 1 - 20 - 3y, = y Tx - S(u, v). y - h(u, v). 26. 1 = x + 4y, v = 3x + 2y 3. 27-30. Double integrals-transformation given To evaluate the fol- Using the transformation T x - u + vy " -v, the image of lowing integrals, carry out these steps. the unit square S - { (u, v): 0 = = = 1,0 = v = 1 } is a region R 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 1 , f(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 Suppose S is the unit cube in the first octant of wyw-space with and v. one vertex at the origin. What is the image of the transformation Compute the Jacobian. Mix - 1/2, y - v/2.z - w/2? d. Change variables and evaluate the new integral. Basic Skills 27. sydA, where R is the square with vertices (0, 0), (1, 1), 5-12. Transforming a square Let S - { (u, v): 0 Su = 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 - 4 - v. the xy-plane under the following transformations. 5. 7:x - 21, y = v/2 28. zydA, where R = ( (x, y) : 0 s x s 2, x sysx + 4); 6. T.x - -u, y = -v use x - 21. y - 4v + 2u. 7. Tex = (u + v) /2,y - (1 - v) /2 29. x 3 Vx + 2y dl, where 8. Tix = 2u + v. y = 2u 9. Tix = 12 - v.y = 2uv R = { ( x. y ) : 0 s x s 2. - x/ 2 = y = 1 - x); use x - 21, y = V - U. 10. Tix = 2uv, y = 1 - v2 11. T. x = u COS TV, y = u sin TV 30. xy dA, where R is bounded by the ellipse 9x7 + 4y? = 36; 12. nix - v sin Tu, y = v Cos Tu use x - 2u,y - 3v. 13-16. Images of regions Find the image R in the xy-plane of the re- 31-36. Double integrals-your choice of transformation Evaluate gion S using the given transformation T. Sketch both R and S. the following integrals using a change of variables. Sketch the original 13. S = { (u, v): v S 1 - 1, u 2 0, v 2 0); nix = u,y = v and new regions of integration, R and S. 14. S = ((u, v): 42 + v. s 1); 1.x = 21,y = 4v 31. [ Vx - y dedy 15. S - { ( 1, v ) : 1 S u s 3, 2 5 v S 4); nix = u/v,y = v 16. S = { (1, v): 2 S u s 3, 3 S v S 6}; Tix = u, y = v/u 32. / Vy - x7 1, where R is the diamond bounded by 17-22. Computing Jacobians Compute the Jacobian J(u, v) for the y - x = 0, y - x = 2, y + x = 0, and y + x = 2 following transformations. 17. nix = 3u, y = -3v 33. /4, where R is the parallelogram bounded 18. nix = 4v,y = -2u by y - x = 1,y - x = 2, y + 2x = 0, and y + 2x = 4 19. nix = 2uv, y = 12- v2 34. " dA, where R is the region bounded by the hyperbolas 20. Tix = u COS TV, y = usin TV 21. Tix = (u + v)/ V2,y = (1 - v)/V2 xy = I and xy = 4, and the lines y/x = 1 and y/x = 3 22. Tix = 1/v, y = V
