hwsol_21

hwsol_21 - Homework Assignment 21 - (13.4) - Solutions Page...

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Homework Assignment 21 - (13.4) - Solutions Page 1051 : 4*, 6*, 8*, 16*, 19* 4. z ! 4 y " 3 x 2 ,0 ! y ! 2 x ! x ! 2 f ! x , y " ! 4 y " 3 x 2 , f x ! 6 x , f y ! 4, 1 " f x 2 " f y 2 ! 1 " 36 x 2 " 16 ! 17 " 36 x 2 S ! " 0 2 " 0 2 x 17 " 36 x 2 dydx ! 2 " 0 2 x 17 " 36 x 2 dx u ! 17 " 36 x 2 , du ! 72 xdx ! 2 " 17 161 u 1 72 du ! 1 36 2 3 u 3/2 17 161 ! 1 54 ! 161 3/2 # 17 3/2 " ! 36. 53274 6. z ! x 2 " y 2 , z ! 4. When z ! 4, x 2 " y 2 ! 4 f ! x , y " ! x 2 " y 2 , f x ! 2 x , f y ! 2 y ,1 " f x 2 " f y 2 ! 1 " 4 x 2 " 4 y 2 S ! " 0 2 ! " 0 2 1 " 4 r 2 rdrd " u ! 1 " 4 r 2 , du ! 8 rdr ! ! 2 ! " " 1 17 u 1 8 du ! ! 4 2 3 u 3/2 1 17 ! ! 6 ! 17 3/2 # 1 " ! 36. 1769 8. z ! x 2 " y 2 , x 2 ! y ! 4. When y ! 4, x 2 ! 4, x !# 2 f ! x , y " ! x 2 " y 2 , f x ! x x 2 " y 2 , f y ! y x 2 " y 2 1 " f x 2 " f y 2 ! 1 " x 2 x 2 " y 2 " y 2 x 2 " y 2 ! 2 S ! " # 2 2 " x 2 4 2 dydx ! " # 2 2 2 ! 4 # x 2 " dx ! 22 4 x # 1 3 x 3 0 2 ! 8 # 8 3 ! 32 2 3 ! 15. 08494 16. z ! x 2 " y 2 , inside r ! 2 # 2cos " , R :0 ! " ! 2 ! ! r ! 2 # " f ! x , y " ! x 2 " y 2 , f x ! 2 x , f y ! 2 y " f x 2 " f y 2 ! 1 " 4 x 2 " 4 y 2 ! 1 " 4 r 2 S ! " 0 2 ! " 0 2 # " 1 " 4 r 2 rdrd " ! 86. 51693 19. z ! 4 # x 2 # y 2 , above z ! 0. When z ! 0, x 2 " y 2 ! 4 f ! x , y " ! 4 # x 2 # y 2 , f x ! # x 4 # x 2 # y 2 , f y !
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