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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

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves.

y = 1 + x 4 ,   y = 3 x 3

To determine

To find:

Approximate x coordinate of the points of intersection of the given curves by using the graph, then use the calculator to find (approximately) the volume of the solid obtained by rotating the bounded region about x axis.

Explanation

1) Concept:

i. If the cross section is a washer with inner radius rin and outer radius rout, then area of the washer is obtained by subtracting the area of the inner disk from the area of the outer disk,

A=π outer radius2 -π inner radius2

ii. The volume of the solid revolution about x-axis is

V= abA(x)dx

2) Given:

y=1+x4, y=3-x3; about the x axis

3) Calculation:

Region bounded by given curves is shown below:

From the above graph,

x coordinate of the points of intersection: x=-1 and  x   0.857

The cross section is perpendicular to the x-axis, and it is a washer.

Thus, the outer radius is the distance from the curve y= 3-x3  to the axis of rotation, x-axis, and the inner radius is the distance from the curve 1+x4 to the axis of rotation, x-axis

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