Could use some help in solving problem 30 in preparation for my midterm

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