Explanation Given information: The radius of cylinder is denoted as R . The radius of hole is denoted as r . The cylinder and the hole are at right angles to the axis of cylinder. Calculation: Consider x -axis to be the axis of cylindrical hole of radius r. Sketch the quarter cross section of the cylinder with hole as shown in Figure 1. Refer to Figure 1. A quarter of the cross-section through y , perpendicular to the y -axis is the rectangle. The dimensions of the rectangle are represented by using Pythagorean Theorem. x 2 = R 2 − y 2 x = R 2 − y 2 z 2 = r 2 − y 2 z = r 2 − y 2 Find the area of the whole section as shown below. A ( y ) = 4 ( x z ) = 4 ( R 2 − y 2 × r 2 − y 2 ) Find the integral for the volume cut out as shown below. V = ∫ − r r A ( y ) d y (1) Substitute 4 ( R 2 − y 2 r 2 − y 2 ) for A ( y ) in Equation (1)

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.2, Problem 69E

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To set up: An integral for the volume cut out.

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Chapter 6.2, Problem 68E

Chapter 6.2, Problem 70E

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Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 7E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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To set up: An integral for the volume cut out.

Solution Summary: The author explains how to set up an integral for the volume cut out.

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To set up: An integral for the volume cut out.

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Chapter 6 Solutions