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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Section
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x 2 about the following lines.

(a) The x -axis

(b) The y -axis

(c) y = 2

To determine

a)

To find:

The volume of the solid rotating the region about x-  axis

Explanation

1) Concept:

Let  S  be a solid that lies between x=a and  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=1nAxi*x=abAxdx 

2) Given:

The equations of the curves are y=x & y=x2.

3) Calculation:

The equations of curves are

y=x & y=x2

Draw the curves about  x  axis.

The curves y=x & y=x2 intersect at the points 0,0&1,1.

The region between them is the solid of rotation and across section perpendicular to x  axis.

The cross section in plane Px has the shape of a washer with the inner radius y=x2 and outer radius y=x.

So, find the cross sectional area by subtracting the area of the outer circle from the area of the inner circle, i.e

To determine

b)

To find:

The volume of a solid rotating the region about y-  axis

To determine

c)

To find:

The volume of the solid rotating the region about y=2.

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