Volume by the Disc/Washer Method. The disc/washer method requires you to visualize a series of discs or washers, compute their volume and add the volumes.

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Disk and Washer Methods

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Volume by the Disc/Washer Method

The disc/washer method requires you to visualize a series of discs or washers, compute their volume and add the volumes of all possible discs/washers. The cross sections of a typical disc/washer would be as shown below.

Rotation of a vertical disc/washer.

Rotation of a horizontal disc/washer

Figure is a cylinder with height, dx, and radius, f(x). Figure is a washer and its volume is the difference between two cylinders.

Figure is a cylinder with height dy and radius, f(y). Figure is a washer and its volume is the difference between two cylinders.

Ex: Find the volume of the solid formed by revolving the region bounded by f(x) from 0 to  around the x axis.

Ex: Find the volume of the solid formed by revolving the region bounded by f(x) and y = 1 around y = 1. Intersections at (-1,1) and (1,1).

Ex: Find the volume of the solid formed by revolving the region bounded by f(x) and g(x) around the x axis.

Ex: Find the volume of the solid formed by revolving the region bounded by the graph of y = x 2, y = 0, x = 0, and x = 1 around the y-axis. Area below y=1 is a cylinder with radius 1 and height 1. Volume is .

Ex. A manufacturer drills a hole through the center of a metal sphere of radius 5 inches. The hole has a radius of 3 inches. What is the volume of the resulting metal ring?

Ex. A pontoon is to be made in the shape formed by revolving the region bounded by f(x) and the x-axis around the x-axis. What is its volume?

4. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

8. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis.

12. Find the volume of the solid generated by revolving the region bounded by the graphs abound the indicated lines. A. y-axis:

B. x-axis C. y = 8.

D. Line x = 2

16. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4.

20. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6.

24. Find the volume of the solid generated by revolving the region bounded by the graphs about the x-axis.

28. Find the volume of the solid generated by revolving the region bounded by the graphs about the x-axis.

41. If the proportion of the line y =.5x lying n the first quadrant is revolved about the x-axis, a cone is generated. Find the volume of the cone extending from x =0 to x = Use the disc method to verify that the volume of a sphere is:

r-r H 45. A cone with base of radius r and height H is cut by a plane parallel to and h units above the base. Find the volume of the solid below the plane (the frustum of the cone). h y H-y x

47. A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of f(x) and the x-axis around the x-axis, where x and y are measured in meters. Find the tanks volume.

49. Find the volume of the solid generated if the upper half of the following ellipse is revolved about: a. The x-axis to form a prolate spheroid. b. The y-axis to form an oblate spheroid a. b.