ISM_T11_C15_C - 968 Chapter 15 Multiple Integrals 15.5...

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968 Chapter 15 Multiple Integrals 15.5 MASSES AND MOMENTS IN THREE DIMENSIONS 1. I y z dx dy dz a y z dy dz a yz dz x c2 b2 a2 œ± œ ± œ ± ''' '' ' Î Î Î Î Î Î ÎÎÎ ÎÎ Î Î Î ab ’“ ## # y 3 $ a bz dz ab z ab b c b c ; œ ± œ ± œ ± œ ± ' Î Î Î Î Š‹’“ Š‹ab bb z b c c a b c M 12 12 3 12 12 1 12 $ #$ # $ # # # # R ; likewise R and R , by symmetry xy z œœ œ ÉÉ É bc ac 12 12 12 ±± ± 2. The plane z is the top of the wedge I y z dz dy dx œÊ œ ± 42 y 3 ² x 324 3 34 4 2 y 3 Î Ð ÑÎ dy dx dx 208; I x œ² ± ± œ œ œ ± ' Î 32 3 3 34 3 2 y 3 y 8y 2y 8(2 y) 3 3 81 81 3 64 104 # $$ ² dy dx 12x dx 280; ± ± œ ± œ ' 3 3 ˆ‰ (4 2y) x (4 2y) 81 3 3 81 3 4x 64 32 ²² # $ # # I x y dz dy dx x y dy dx 12 x 2 dx 360 z 3 3 2 y 3 3 œ ± ² œ ± œ ' Î a b # 8 33 2y 3. I y cy dy dx dx x 000 00 0 abc a œ ± œ ± œ ' Š‹ Š # ± cc b c b 3 3 abc b c $ b c where M abc; I a c and I a b , by symmetry œ MM M 3 yz 4. (a) M dz dy dx (1 x y) dy dx x dx ; ² ² œ ² ± œ '' ' ' 0 0 11 x1 x y x 1 x 6 # "" M x dz dy dx x(1 x y) dy dx x 2x x dx yz 0 0 x y x 1 ² ² œ ² ± œ ' # $# 24 x y z , by symmetry; I y Êœœœ œ ± " 4 x 0 x y y xy y dy dx (1 x) dx I I , by symmetry ² ± œ ² œ Ê œ œ ' 0 x 1 yx $ % " ( 1xy ) 3 6 30 30 $ (b) R 0.4472; the distance from the centroid to the x-axis is 0 x œ ¸ ± ± œ œ É É I M5 5 16 16 8 4 5 2 x " " # È È 0.3536 ¸ 5. M 4 dz dy dx 4 4 4y dy dx 16 dx ; M 4 z dz dy dx ² œ œ œ ' 004 y 0 y 114 1 xy # # # 23 2 2 16 16y dy dx dx z , and x y 0, by symmetry; œ²œ œ Ê œ œ œ ' 0 1 % 128 128 12 555 I 4 y 4 4y 4y dy dx 4 dx ; x y 0 1 œ ± ² ± œ œ ' # # % 64 1976 7904 3 3 105 105 64y ' I 4 x 4 4x 4x y dy dx 4 x dx y y 0 1 œ ± ² ± œ ± ' # ˆ # # # # 64 8 128 3 7 64y ' ; I 4 x 16 x x y y y dy dx ± œ ² ± ² 4832 63 z y # a b ##% 16 dx œ ' 0 1 2x 2 256 31 5 4 5 # 6. (a) M dz dy dx (2 x) dy dx (2 x) 4 x dx 4 ; ² œ ² ² œ ' ' Î Î Î 24 x 2 0 x 2 2 x 2 2 x x 2 2 ÈÈ È # 1 M x dz dy dx x(2 x) dy dx x(2 x) 4 x dx 2 ; yz x 2 0 x 2 2 x 2 2 x x 2 2 ² œ ² ² œ ² ' ' Î Î Î È # 1 M y dz dy dx y(2 x) dy dx xz x 2 0 x 2 x 2 2 x x 2 ² ' Î Î Î (2 x) dx 0 x and y 0 œ²² œ Ê œ ² œ ² " ' 2 2 4x 44
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Section 15.5 Masses and Moments in Three Dimensions 969 (b) M z dz dy dx (2 x) dy dx (2 x) 4 x dx xy 24 x 2 0 x 2 2 x 2 2 x x 2 2 œœ ± œ ± ± '' ' ' Î Î Î Î Š‹ ÈÈ # # # # "" ## # È 5 z œÊ œ 1 5 4 7. (a) M 4 dz dy dx 4 r dz dr d 4 4r r dr d 4 4 d 8 ; œ ± œ œ ' ' '' ' ' ' 00 x y 0 0r 0 0 0 2 4 x 4 224 22 2 È ÎÎ Î # # 11 1 )) ) 1 ab $ M zr dz dr d 16 d z , and x y 0, xy r 0 4 2 ± œ œ Ê œ œ œ ''' ' 1 # ) r3 2 6 4 8 33 3 # % 1 by symmetry (b) M 8 4 r dz dr d cr d c 8 c 2 2, œÊœ œ ± œ œ Ê œ ) ) ) '' ' ' r 0 0 0 2c c 2 c 2 1 # È $# # cc 4 1 since c 0 ² 8. M 8; M z dz dy dx dy dx 0; M x dz dy dx œ xy yz 13 1 1 151 15 ’“ z 2 # " ±" 2 x dy dx 4 x dx 0; M y dz dy dx 2 y dy dx 16 dx 32 œ œ œ œ œ ' ' 1 1 1 1 1 xz x 0, y 4, z 0; I y z dz dy dx 2y dy dx 100 dx ; Êœ œ œ œ ³ œ ³ œ œ x 1 1 1 ' ˆ‰ # 2 2 400 3 I x 2x dy dx 3x 1 dx ; y 1 1 1 œ³ œ ³ œ ³ œ ' a b # # 1 6 3 I x y dz dy dx 2 x y dy dx 2 2x dx R R z xz 1 1 1 œ ³ œ ³ œ Ê œ œ ' É # 98 400 50 3 and R y œ É 2 3 9. The plane y 2z 2 is the top of the wedge I (y 6) ³œ Êœ ± ³ L 221 24 2 y 2 Ð ÑÎ cd dy dx; let t 2 y I 4 5t 16t dt 1386; ³ œ ± Ê œ ³ ³ ³ œ ' 2 4 L Š (y 6) (4 y) (2 y) 24 3 24 3 13t 49 ±± ± # " # # $ $ M (3)(6)(4) 36 R Ê œ œ " L É É I
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ISM_T11_C15_C - 968 Chapter 15 Multiple Integrals 15.5...

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