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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Solve Example 9 taking cross-sections to be parallel to the line of intersection of the two planes.

To determine

To find:

Volume of the wedge

Explanation

1) Concept:

i) Let S be a solid that lies between x=a to x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x),  where A is continuous function, then the volume of S is

V=limni=1nA(xi*)x=abA(x)dx

2) Calculation:

Given that the cross section is parallel to the line of intersections of two planes

So, cross section forms the rectangle

Equation of circle of radius r centered at origin is x2+y2=r2

Therefore, x=16-y2

AB is the length of the rectangle

AB=2x

2x=216-y2

To find the width, consider the triangle as shown below,

From the triangle

tan30=BCy

13=BCy

BC=y3=w

Therefore, area of rectangle is Ay=l·w

Ay=216-y2·y3

Ay=23y16-y2

The range of y is 0 to 4

V=abA(y)dy=0423y16-y2dy

V=04y16-y2dy

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