Calculus

# Calculus - Chapter 13 Multiple Integrals Chapter 13 Section...

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Chapter 13 Multiple Integrals 404 Chapter 13 Section 13.1 5. 2 ( , ) 2 , 0 2, 1 1, 4 fxy x y x y n =+ ≤≤ −≤ ≤ = The centers of the four squares are 11 1 1 31 3 1 , , , , and , . 22 2 2 22 2 2  −−   Since the four squares are the same size, 1, i A ∆= for each i . 4 1 22 (,) 11 1 1 31 , (1) , , 2 2 ,( 1 ) 1 1 3 1 2 2 2 2 112 2 6 ii i i Vf u v A ff f f = ≈∆     =−+ +−         +  ++   =++ + = 7. 2 2 1 16 x y x y n = The centers of the sixteen squares are 13 11 1 1 1 3 33 ,, , , , , 44 44 4 4 4 4 44 31 3 1 3 3 53 51 , , , , 44 51 53 7 3 7 1 71 , , , 73 and , . Since the sixteen squares are the same, 1 , 4 i A for each i . 16 1 131 111 7 , 444 4 1331 5 7 71 91 9 2 5 2 3 48 88 8 8 8 8 8 8 8 8 8 8 8 8 8 13 2 i i u v A f = =− + + = +++ + +++ + + + + + + + + =

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Chapter 13 Multiple Integrals 405 9. ( , ) 3 , 1 1, 0 4, 4 fxy x y x y n =−− ≤≤ ≤≤ = The upper right corners of the four rectangles are (0, 2), (0, 4), (1, 2), and (1, 4). Since the rectangles are the same, 2, i A ∆= for each i . 4 1 (,) (0,2)(2) (0,4)(2) (1,2)(2) (1,4)(2) 2( 2 4 1 1) 12 ii i Vf u v ffff ι = ≈∆ Α =++ + =−−+− =− 11. 3 , 0 4, 0 2 x y =− ≤≤ ≤≤ The centers of the areas are 11 13 51 53 ,, , a n d ,. 22          The areas are 12 34 1, 3. AA AA == 4 1 ,( 1 ) 1 ) 3 ) 3 ) 11 01 7 3 6 3 40 i i u v A = =+++ =⋅+⋅+⋅+⋅ = 13. 21 01 2 1 1 0 2 2 23 0 0 (2 ) ) [] 2 2 3 16 3 R y y xy d A d y d x xy y d x xd x x = −=    = ∫∫ ∫ ∫ 15. 41 20 4 1 2 0 2 4 2 4 2 2 2 44 [2 ] 2( 1 ) ( 1 ) [ ] 12( 1) yy R y y y xe dA xe dydx xe dx xe dx e x e = = = = = 17. 32 00 3 2 0 0 3 2 0 3 2 0 6 (1 ) ) (3 ) 1 3 2 19 1 xy xy R xy x x y y ye dA ye dxdy d y ed y ye e = = 19. –2 –1 0 1 2 0 5 10 –2 –1 0 1 2 21. –2 0 2 4 0 5 10 –2 –1 0 1 2 23. 1 2 2 0 0 1 1 0 0 ) [ ] 62 [ ] 2 x yx y d y d x x y y d x x x = = += + =
Chapter 13 Multiple Integrals 406 25. 4 24 2 2 00 0 0 2 2 0 2 3 0 1 (2 ) 2 2 16 16 128 33 yy y x xy d x d y x x y d y yd y y = =  +=+   = == ∫∫ 27. 12 1 2 2 0 0 1 5/2 2 0 1 7/2 3 0 (4 ) [2 ] (8 2 ) 16 2 73 62 21 y x xy yd x d y x y x y d y d y = = += + =+ = 29. 2 2 2 2 2 22 2 0 0 2 0 2 4 0 2 1 y x y y ed x d y x e d y ye dy ee = = = = 31. 1/ 41 / 4 10 1 0 4 1 4 1 1 cos sin 1 sin1 sin1 ln sin1(ln 4) 2ln2(sin1) yx x y xydydx xy dx x dx x x = = = = = = = 33. 2 1 0 0 1 1 34 0 0 1 2 2 1 2 x y xd x d x xdx x = = = = 2 1 23 0 0 1 3 0 0 1 3 8 3 [] y x y x d y y y = = = = Therefore, . x y 35. 4 3 2 3 01 0 1 3 2 0 0 1 () 3 (3 21) 1 ] 90 y y x y dydx x y y dx x xx = = + = 37. On the xy -plane, the region R lies between the parabola 2 = and the line y = 1. Thus, 2 11 , 1 .

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## This homework help was uploaded on 04/19/2008 for the course MATH 105 taught by Professor Schroder during the Spring '08 term at Millersville.

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Calculus - Chapter 13 Multiple Integrals Chapter 13 Section...

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