1. Volume by Parallel Cross Section; Disks and Washers
Figure shows a plane region Ω and a solid formed by translating Ω along a line perpendicular to the plane of Ω. Such a solid is called a right cylinder with cross section Ω.
If Ω has area A and the solid has height h, then the volume of the solid is a simple product:
V = A · h (cross-sectional area · height)
Salas, Hille, Etgen Calculus: One and Several Variables
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2. Volume by Parallel Cross Section; Disks and Washers
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

3. Volume by Parallel Cross Section; Disks and Washers
If the cross-sectional area A(x) varies continuously with x, then we can find the
volume V of the solid by integrating A(x) from x = a to x = b:
Salas, Hille, Etgen Calculus: One and Several Variables
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4. Volume by Parallel Cross Section; Disks and Washers
Example 1 Find the volume of the pyramid of height h given that the base of the pyramid is a square with sides of length r and the apex of the pyramid lies directly above the center of the base.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

5. Volume by Parallel Cross Section; Disks and Washers
Example 2
The base of a solid is the region enclosed by the ellipse
Find the volume of the solid given that each cross section perpendicular to the x-axis is and isosceles triangle with base in the region and altitude equal to one-half the base.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

6. Volume by Parallel Cross Section; Disks and Washers
Example 3 The base of a solid is the region between the parabolas
x = y and x = 3 – 2y 2.
Find the volume of the solid given that the cross sections perpendicular to the x-axis are squares.
Salas, Hille, Etgen Calculus: One and Several Variables
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7. Volume by Parallel Cross Section; Disks and Washers
Solids of Revolution: Disk Method
The volume of this solid is given by the formula
Salas, Hille, Etgen Calculus: One and Several Variables
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8. Volume by Parallel Cross Section; Disks and Washers
Example 4 Find the volume of a circular cone of base radius r and height h.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

9. Volume by Parallel Cross Section; Disks and Washers
Example 5 Find the volume of a sphere of radius r.
Salas, Hille, Etgen Calculus: One and Several Variables
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10. Volume by Parallel Cross Section; Disks and Washers
We can interchange the roles played by x and y. By revolving about the y-axis the
region of Figure , we obtain a solid of cross-sectional area A(y) = π[g(y)]2 and
volume
Salas, Hille, Etgen Calculus: One and Several Variables
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11. Volume by Parallel Cross Section; Disks and Washers
Example 6 Let Ω be the region bounded below by the curve y = x2/3 + 1, bounded to the left by the y-axis, and bounded above by the line y = 5. Find the volume of the solid generated by revolving Ω about the y-axis.
Salas, Hille, Etgen Calculus: One and Several Variables
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12. Volume by Parallel Cross Section; Disks and Washers
Solids of Revolution: Washer Method
The washer method is a slight generalization of the disk method. Suppose that f and g
are nonnegative continuous functions with g(x) ≤ f (x) for all x in [a, b]. If we revolve the region Ω about the x-axis, we obtain a solid. The volume of this solid is given by the formula
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

13. Volume by Parallel Cross Section; Disks and Washers
Example 7 Find the volume of the solid generated by revolving the region between
y = x and y = 2x
about the x-axis.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

14. Volume by Parallel Cross Section; Disks and Washers
As before, we can interchange the roles played by x and y.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

15. Volume by Parallel Cross Section; Disks and Washers
Example 7 Find the volume of the solid generated by revolving the region between
y = x2 and y = 2x
about the y-axis.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

16. Volume by the Shell Method
Volume of the cylindrical shell in Figure is given by
Salas, Hille, Etgen Calculus: One and Several Variables
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17. Volume by the Shell Method
Salas, Hille, Etgen Calculus: One and Several Variables
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18. Volume by the Shell Method
Example 1 The region bounded by the graph of f (x) = 4x – x2 and the x-axis from x = 1 to x = 4 is revolved about the y-axis. Find the volume of the resulting solid.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

19. Volume by the Shell Method
The volume generated by revolving Ω about the y-axis is given by the formula
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

20. Volume by the Shell Method
Example 2 Find the volume of the solid generated by revolving the region between
y = x and y = 2x
about the y-axis.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

21. Volume by the Shell Method
The volume generated by revolving Ω about the x-axis is given by the formula
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

22. Volume by the Shell Method
Example 2
Find the volume of the solid generated by revolving the region between
y = x and y = 2x
about the x-axis.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

23. Volume by the Shell Method
Example 3 A round hole of radius r is drilled through the center of a half-ball of radius a (r＜a). Find the volume of the remaining solid.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

24. Volume by the Shell Method
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

25. Volume by the Shell Method
Example 4
is revolved about the line x = -2. Find the volume of the solid which is generated.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.