Questions 22 and 27 please

2.1 Double Integrals over Rectangular Regions 117 13 . J J aredy axdy 14 . J . J. 14 x dxdy 30 . y' sinxy ? d ; R = { ( x, y ) : 0 5 x 5 2, 0 5 y s VT/2} 15 . fins erty dady 16 . J. " J rsocodr de 31. / (1 4 x ) 2 dA ; R = { ( xy):0 5 x 54, 1 5 y =2) 17-25. Double integrals Evaluate each double integral over the 32-34. Average value Compute the average value of the following region R by converting it to an iterated integral. functions over the region R. 17 . / / ( x + 2y) dA ; R = { ( x, > ) : 0 5 x 5 3, 1 5 y s4) 32. f ( x, y ) = 4 - x - y; R = { ( x, y): 0 s x = 2, 0 sys 2} 33. f ( x, y ) = ey; R = { ( x, y ) : 0 = x s 6, 0 s y s In2} . // ( x2 + xy ) dA ; R = ( ( x, y ) : 1 5 x 52, - 1 sys 1} 34. f( x, y) = sin x sin y; R = {(x, y) : 0 s x ST, 0 s y ST} 35-36. Average value 35. Find the average squared distance between the points of 19 . 4x 3 cos y dA; R = { ( x, y) : 1 5 x 5 2, 0 5 y s m/2) R = { (x, y): -2 5 x $ 2, 0 = y $ 2} and the origin. 36. Find the average squared distance between the points of R = {(x, y): 0 S x ): 0 5 x51, 1sys4) statements are true and give an explanation or counterexample. 21. a. The region of integration for J J, 4 dx dy is a square. b. If f is continuous on R2, then 22 . xy sin x 2 dA ; R = { ( x , y ) : 0 s x s VT / 2, 0 = y = 1} [ Sif(x, y ) dx dy = S. Sif (x y) dy dx. c. If f is continuous on R2, then S . S if(x, y ) dx dy = Si Sif (x, y) dy dx. 23. ex + 2y dA; R = { (x, y): 0 = x s In2, 1 s y s In 3} 38. Symmetry Evaluate the following integrals using symmetry arguments. Let R = { (x, y) : - a s xsa, -b sys b}, wh a and b are positive real numbers. 24 . ( x 2 - 12 ) 2 dA ; R = { ( x , y ) : - 1 5 x = 2, 0 = y = 1} a . sin (x - y) J ( 5 - 35 ) dA; R = { ( x, y): 05x51, -1sys1} b. 25. x 2 + y ? + 1 39. Computing populations The population densities in nine dist 26-31. Choose a convenient order When converted to an iterated of a rectangular county are shown in the figure. integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. y (mi) A Population densities have units of people/mi-. 26 . y cos xy dA ; R = ( ( x, y ): 0 = x 5 1, 0 = y ST/ 3} 250 200 150 ./ / (+ Debt ) da ; R = { ( zy ): 0 5x5 1 , - 1sys1} 500 400 250 350 300 150 28. x sec2 xy dA; R = { ( x, y): 0 5 x S T / 3, 0 s y s 1} x (mi) No / w 29. Ox' edA ; R = { ( x, y ) : 0 5 x = 2, 0sys2} a. Use the fact that population = (population density) X to estimate the population of the county. b. Explain how the calculation of part (a) is related to Riem sums, and double integrals