16.3.Ex37

# 16.3.Ex37 - 930 C H A P T E R 16 M U LTI P L E I N T E G R...

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930 CHAPTER 16 MULTIPLE INTEGRATION (ET CHAPTER 15) The centroid of the tetrahedron is thus P = ( 1 . 5 , 1 , 2 ) . 36. Find the centroid of the region described in Exercise 30. SOLUTION The region W is bounded by the cylinders z = 1 y 2 and y = x 2 , and the plane y = 0and y = 1. In Exercise 30 we showed that V = Volume ( W ) = 16 21 and W can be described by the following inequalities: 1 x 1 , x 2 y 1 , 0 z 1 y 2 We use iterated integrals to compute the coordinates x , y ,and z of the centroid. We get x = 1 V ZZZ W xdV = 21 16 Z 1 1 Z 1 x 2 Z 1 y 2 0 xdzdydx = 21 16 Z 1 1 Z 1 x 2 xz ¯ ¯ ¯ ¯ 1 y 2 z = 0 dydx = 21 16 Z 1 1 Z 1 x 2 x ³ 1 y 2 ´ = 21 16 Z 1 1 x Ã y y 3 3 ± ¯ ¯ ¯ ¯ 1 y = x 2 dx = 21 16 Z 1 1 x Ã 1 1 3 Ã x 2 x 6 3 ±± = 21 16 Z 1 1 Ã x 7 3 x 3 + 2 x 3 ± = 0 y = 1 V W ydV = 21 16 Z 1 1 Z 1 x 2 Z 1 y 2 0 ydzdydx = 21 16 Z 1 1 Z 1 x 2 yz ¯ ¯ ¯ ¯ 1 y 2 z = 0 = 21 16 Z 1 1 Z 1 x 2 y ³ 1 y 2 ´ = 21 16 Z 1 1 Z 1 x 2 ³ y y 3 ´ = 21 16 Z 1 1 Ã y 2 2 y 4 4 ± ¯ ¯ ¯ ¯ 1 y = x 2 = 21 16 Z 1 1 Ã 1 2 1 4 Ã x 4 2 x 8 4 ±± = 21 8 Z 1 0 Ã x 8 4 x 4 2 + 1 4 ± = 21 8 Ã x 9 36 x 5 10 + x 4 ± ¯ ¯ ¯ ¯ 1 0 = 21 8 µ 1 36 1 10 + 1 4 ² = 7 15 z = 1 V W zdV = 21 16 Z 1 1 Z 1 x 2 Z 1 y 2 0 zdzdydx = 21 16 Z 1 1 Z 1 x 2 z 2 2 ¯ ¯ ¯ ¯ 1 y 2 z = 0 = 21 16 Z 1 1 Z 1 x 2 ³ 1

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16.3.Ex37 - 930 C H A P T E R 16 M U LTI P L E I N T E G R...

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