16.3.Ex16-17

16.3.Ex16-17 - 908 C H A P T E R 16 M U LTI P L E I N T E G...

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908 CHAPTER 16 MULTIPLE INTEGRATION (ET CHAPTER 15) = Z 1 0 Ã 13 3 y 2 2 ± dy = 13 3 y y 3 6 ¯ ¯ ¯ ¯ 1 0 = 13 3 1 6 = 25 6 = 4 1 6 16. Integrate f ( x , y , z ) = z over the region W below the upper hemisphere of radius 3 as in Figure 12, but lying over the triangle in the xy -plane bounded by the lines x = 1, y = 0, and x = y . x y z 1 W 1 D 3 3 x 2 + y 2 + z 2 = 9 FIGURE 12 SOLUTION x y z 1 1 3 3 The upper surface is the hemisphere z = p 9 x 2 y 2 and the lower surface is the -plane z = 0. The projection of V onto the -plane is the triangle D shown in the ±gure. y y = x x 1 1 0 D We compute the triple integral as the following iterated integral: ZZZ V zdV = ZZ D Ã Z 9 x 2 y 2 0 zdz ± dA = D z 2 2 ¯ ¯ ¯ ¯ 9 x 2 y 2 0 = D 9 x 2 y 2 2 = Z 1 0 Ã Z x 0 9 x 2 y 2 2 ± dx = Z 1 0 9 y x 2 y y 3 3 2 ¯ ¯ ¯ ¯ x y = 0 = Z 1 0 Ã 9 x 2 2 x 3 3 ± = 9 x 2 4 x 4 6 ¯ ¯ ¯ ¯ 1 0 = 2 1 12 17. Calculate the integral of f ( x , y , z ) = e x + y + z over the tetrahedron W in Figure 13.

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SECTION 16.3 Triple Integrals (ET Section 15.3) 909
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16.3.Ex16-17 - 908 C H A P T E R 16 M U LTI P L E I N T E G...

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