Problems 15,28,32 please
2.7 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 i = xy and v =y/x (or vice versa). The new region of integration is the rectangle S = { (u, v): 1 5 1 5 3. I s v 3 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 irv-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 - 2x -y 24. u = xy, V = x 2. Explain how to compute the Jacobian of the transformation Tax - s(u, v), y = h(u, v). 25. 1 2x - 3y.V y-x 26. u = x + 4y, v = 3x + 2 3. Using the transformation Tix - u + vy = u - v. the image of 27-30. Double Integrals- transformation given To evaluate the fol- the unit square S - { (u, v): O s us .0 5 vs ) is a region lowing integrals, carry out these steps. R in the xy-plane. Explain how to change variables in the integral 1. Sketch the original region of Integration R in the xy-plane and the Jef(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 uvw-space with and v. one vertex at the origin. What is the image of the transformation c. Compute the Jacobian. Tex - 1/2, y = v/2, 2 - w/2? d. Change variables and evaluate the new integral. Basic Skills 27. xy dA, where R is the square with vertices (0, 0), (1, 1), 5-12. Transforming a square Let S = {(u, v): O S us 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 = a + v.y - u- the xy-plane under the following transformations. 5. Tix = 21. > = v/2 28. xhy dA, where R = ( ( x, y ): 0 s x 3 2, x 5 y s x + 4); 6. Tix = -u, y = -v use x = 211, y = 4v + 2u. 7. Tex = (u + v)/2, y = (1 - v)/2 29. x2 Vx + 2y dA, where 8. nix = 21 + v,y = 21 9. Tix = 1 - v, y = 2uv R = { (x, y): 0 s x s 2, -x/2 5 y s l - x}; use x = 21, y = v - U. 10. Tix = 2uv, y = 12- v2 11. Tix = u COs Try, y = u sin TV . xy dA, where R is bounded by the ellipse 9x2 + 4y? = 36; 12. Tix = v sin wu, 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- 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 = 1 - 1, u 2 0, v 2 0); nix = u, y = v2 and new regions of integration, R and S. 14. S = { (u, v) : 12 + v. s 1 }; nix = 21, y = 4v 31. [ vx- yardy 15. S = { (u, v): 1 S u s 3, 2 5 v s 4); Tix = u/ v, y = v 16. S = { (u, v): 2 S u s 3, 3 5 v S 6); Tix = u,y = v/u 32. Vy7 - x- dl, where R is the diamond bounded by 17-22. Computing Jacobians Compute the Jacobian J(u, v) for the - x =0,y - x = 2, y + x = 0, and y + x = 2 following transformations. y + 2r + 1 ) dA, where R is the parallelogram bounded 17. nix = 3u, y = -3v 18. nix = 4v, y = -2u y - x = 1.y - x = 2, y + 2r = 0, and y + 2x = 4 19. T. x = 2uv, y = 12- v2 34. edA, where R is the region bounded by the hyperbolas 20. 7: x - u COS TV, y = 1I sin TV 21. Tix = (u + V)/ V2,y = (u- v)/V2 xy = I and xy = 4, and the lines y/x = 1 and y/x = 3 22. Tix = u / vy = V
