Explanation Given : The region bounded by z = 4 − y 2 , z = 0 , y = 2 x , x = 0 Calculation : Step 1: In Example 5 we considered the triple integral ∭ W x y z 2 d V , where W is the region bounded by z = 4 − y 2 , z = 0 , y = 2 x , x = 0 . We now write the triple integral as an iterated integral in the orders dz dx dy , dx dy dz , and dy dx dz dz dx dy : The upper surface z = 4 − y 2 projects onto the xy -plane on the triangle defined by the lines y = 2 , y = 2 x and x = 0 Step 2: We express the line y = 2 x as x = y 2 and write the triple integral as ∭ W x y z 2 d V = ∭ D ( ∫ 0 4 − y 2 x y z 2 d z ) d A = ∫ 0 2 ∫ 0 y 2 ∫ 0 4 − y 2 x y z 2 d z d x d y Step 3: dx dy dz : The projection of W onto the yz -plane is the domain T (see Example 5) defined by the inequalities T : 0 ≤ y ≤ 2 , 0 ≤ z ≤ 4 − y 2 This region can also be expressed as 0 ≤ z ≤ 4 , 0 ≤ y ≤ 4 − z Step 4: As explained in Example 5, the region W consists of all points lying between T and the “left-face” y = 2 x , or x = y 2 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.3, Problem 27E

To determine

To calculate: The expression of triple integral $\int \int \int_{W} x y z^{2} d V$ in the three orders dzdxdy, dxdydz and dydxdz. where W is the region bounded by $z = 4 - y^{2}, z = 0, y = 2 x, x = 0$

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Chapter 16.3, Problem 26E

Chapter 16.3, Problem 28E

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Chapter 16 Solutions

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The expression of triple integral ∫ ∫ ∫ W x y z 2 d V in the three orders dzdxdy, dxdydz and dydxdz. where W is the region bounded by z = 4 − y 2 , z = 0 , y = 2 x , x = 0

Solution Summary: The author calculates the expression of the triple integral, W x y z 2 d V, where W is the region bounded by Z = 0.

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To calculate: The expression of triple integral $\int \int \int_{W} x y z^{2} d V$ in the three orders dzdxdy, dxdydz and dydxdz. where W is the region bounded by $z = 4 - y^{2}, z = 0, y = 2 x, x = 0$

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Chapter 16 Solutions

The expression of triple integral ∫ ∫ ∫ W x y z 2 d V in the three orders dzdxdy, dxdydz and dydxdz. where W is the region bounded by z = 4 − y 2 , z = 0 , y = 2 x , x = 0

Solution Summary: The author calculates the expression of the triple integral, W x y z 2 d V, where W is the region bounded by Z = 0.

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To calculate: The expression of triple integral ∫∫∫Wxyz2dV in the three orders dzdxdy, dxdydz and dydxdz. where W is the region bounded by z=4−y2,z=0,y=2x,x=0

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Chapter 16 Solutions

To calculate: The expression of triple integral $\int \int \int_{W} x y z^{2} d V$ in the three orders dzdxdy, dxdydz and dydxdz. where W is the region bounded by $z = 4 - y^{2}, z = 0, y = 2 x, x = 0$