Volume by Parallel Cross Section; Disks and Washers Figure 6.2.1 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 Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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 Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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.

Volume by Parallel Cross Section; Disks and Washers Example 3 The base of a solid is the region between the parabolas x = y 2 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 Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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 Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 6.2.10, we obtain a solid of cross-sectional area A(y) = π[g(y)]2 and volume Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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 x-axis. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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.

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.

Volume by the Shell Method Volume of the cylindrical shell in Figure 6.3.2 is given by Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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

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.

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.

Volume by the Shell Method Example 2 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.

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.

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

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.

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

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.