Explanation Given : The wedge-shaped region contained in the cylinder x 2 + y 2 = 9 over the bounded above by the plane z = x and below by the xy plane Formula Used : Change of Variables Calculation : Step 1: Express W in cylindrical coordinates. W is bounded above by the plane z = x v and below by z = 0 , therefore 0 ≤ z ≤ x , in particular x ≥ 0 . Hence, W projects onto the semicircle D in the xy-plane of radius 3, where x ≥ 0. D: − π 2 ≤ θ ≤ π 2 , 0 ≤ r ≤ 3 The upper surface is z = x = r c o s θ and the lower surface is z = 0 . Therefore, W: − π 2 ≤ θ ≤ π 2 , 0 ≤ r ≤ 3 , 0 ≤ z ≤ r c o s θ Step 2 : Set up an integral in cylindrical coordinates and evaluate. The volume of W is the triple integral ∭ W 1 d V ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.4, Problem 21E

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To calculate : The volume of the wedge-shaped region contained in the cylinder $x^{2} + y^{2} = 9$ over the bounded above by the plane $z = x$ and below by the xy plane.

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Chapter 16.4, Problem 20E

Chapter 16.4, Problem 22E

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Chapter 16 Solutions

Calculus - Standalone book

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The volume of the wedge-shaped region contained in the cylinder x 2 + y 2 = 9 over the bounded above by the plane z = x and below by the xy plane.

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To calculate : The volume of the wedge-shaped region contained in the cylinder x2+y2=9 over the bounded above by the plane z=x and below by the xy plane.

Want to see the full answer?

Chapter 16 Solutions

To calculate : The volume of the wedge-shaped region contained in the cylinder $x^{2} + y^{2} = 9$ over the bounded above by the plane $z = x$ and below by the xy plane.