Explanation Given : W is the domain bounded by the elliptic cylinder x 2 4 + y 2 9 = 1 and the sphere x 2 + y 2 + z 2 = 16 in the first octant x ≥ 0 , y ≥ 0 , z ≥ 0 Calculation : Step 1: The upper surface is the sphere x 2 + y 2 + z 2 = 16 , or z = 16 − x 2 − y 2 , and the lower surface is the xy -plane, z = 0 . Step 2: The projection of W onto the xy -plane is the region in the first quadrant bounded by the ellipse ( x 2 ) 2 + ( y 3 ) 2 = 1 , or y = 3 2 4 − x 2 Therefore, the triple integral over W is equal to the following iterated integral: ∭ W x z d V = ∬ D ( ∫ 0 16 − x 2 − y 2 x z d z ) d A = ∬ D 1 2 x z 2 | z = 0 16 − x 2 − y 2 d A = 1 2 ∬ D x ( 16 − x 2 − y 2 ) d A = 1 2 ∫ 0 2 ∫ 0 3 2 4 − x 2 16 x − x 3 − x y 2 d y d x = 1 2 ∫ 0 2 16 x y − x 3 y − 1 3 x y 3 | z = 0 3 2 4 − x 2 = 1 2 ∫ 0 2 24 4 − x 2 − 3 2 x 3 4 − x 2 − 9 8 x ( 4 − x 2 ) 3 2 d x = 12 ∫ 0 2 x 4 − x 2 d x − 3 4 ∫ 0 2 x 3 4 − x 2 d x − 9 16 ∫ 0 2 x ( 4 − x 2 ) 3 2 d x Step 3: The first and third integrals are u-substitution problems, which is very simple...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 16.3, Problem 23E

To determine

To calculate: The value of $\int \int \int_{W} x z d V$ where W is the domain bounded by the elliptic cylinder $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ and the sphere $x^{2} + y^{2} + z^{2} = 16$ in the first octant $x \geq 0, y \geq 0, z \geq 0$

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Chapter 16.3, Problem 22E

Chapter 16.3, Problem 24E

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Chapter 16 Solutions

Calculus - Standalone book

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The value of ∫ ∫ ∫ W x z d V where W is the domain bounded by the elliptic cylinder x 2 4 + y 2 9 = 1 and the sphere x 2 + y 2 + z 2 = 16 in the first octant x ≥ 0 , y ≥ 0 , z ≥ 0

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To calculate: The value of ∫∫∫WxzdV where W is the domain bounded by the elliptic cylinder x24+y29=1 and the sphere x2+y2+z2=16 in the first octant x≥0,y≥0,z≥0

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Chapter 16 Solutions

To calculate: The value of $\int \int \int_{W} x z d V$ where W is the domain bounded by the elliptic cylinder $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ and the sphere $x^{2} + y^{2} + z^{2} = 16$ in the first octant $x \geq 0, y \geq 0, z \geq 0$