Explanation Given information: A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of cylinder. The other plane intersects the first at an angle of 30 ° along a diameter of cylinder. Cross-sections to be parallel to the line of intersection of the two planes. Calculation: Consider the Equation of circle x 2 + y 2 = r 2 The region lies between a = 0 and b = 4 . Substitute 4 for r in the Equation. x 2 + y 2 = 4 2 x 2 + y 2 = 16 Consider the cross-sections perpendicular to the y -axis are rectangles. Sketch the cross section as shown in Figure 1. Refer to Figure 1. The base length of the rectangle as 2 x and height of the rectangle as h . Show the expression for base length: b = 2 x = 2 y 2 − 16 Show the expression for height. h = y tan 30 ° = y 3 The expression to find the volume of the wedge as shown below. V = ∫ a b A ( y ) d y (1) Find the volume of the rectangle as shown below. A ( y ) = 2 x × h = 2 16 − y 2 × y 3 = 2 y 3 16 − y 2 Substitute 0 for a , 4 for b , and 2 y 3 16 − y 2 for A ( y ) in Equation (1). V = ∫ 0 4 2 y 3 16 − y 2 d y = 2 3 ∫ 0 4 y 16 − y 2 d y (2) Consider u = 16 − y 2 (3) Differentiate both side of the Equation (3)

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Chapter 6.2, Problem 77E

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To calculate: The volume of the wedge.

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Chapter 6.2, Problem 76E

Chapter 6.2, Problem 78E

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The volume of the wedge.

Solution Summary: The author explains how a circular cylinder of radius 4 is cut out by two planes, one perpendicular to the axis and the other at an angle of 30°.

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To calculate: The volume of the wedge.

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