Volume by Parallel Cross Section;Disks and Washers Figure 6.2.1 shows a plane region and a solid formed by translating along a line perpendicular to the plane of Such a solid is called a right cylinder with cross section Figure 6.2.1 If has area 4 and the solid has height h,then the volume of the solid is a simple product: V=A·h (cross-sectional area height) Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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 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)
Volume by Parallel Cross Section:Disks and Washers Two elementary examples are given in Figure 6.2.2. V=(r2)h=(cross-sectional area).height =(1.w).h=(cross-sectional area).height Figure 6.2.2 Saks,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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
Volume by Parallel Cross Section:Disks and Washers If the cross-sectional area 4(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: (6.2.1) A(x)dx. area A(x) b xo=a xi-1 Xi xn=b Figure 6.2.3 Figure 6.2.4 Saks,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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:
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. Figure 6.2.5 Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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
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-halfthe base. x=-4 x=a Figure 6.2.6 Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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 1. 2 2 2 2 + = b y a x 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
Volume by Parallel Cross Section;Disks and Washers Example 3 The base of a solid is the region between the parabolas x=y2 and x=3-2y2. Find the volume of the solid given that the cross sections perpendicular to the x-axis are squares. x=3-2y2 x=0x=1 x=3 Figure 6.2.7 Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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
Volume by Parallel Cross Section;Disks and Washers Solids of Revolution:Disk Method The volume of this solid is given by the formula (6.2.3) v= π[fx)Pdx. f() Figure 6.2.8 Sals,Hille,Etgen Calculus One and Several Vaibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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
Volume by Parallel Cross Section:Disks and Washers Example 4 Find the volume of a circular cone of base radius r and height h. (h, V= Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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
Volume by Parallel Cross Section:Disks and Washers Example 5 Find the volume of a sphere of radius r. Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Ine.All rights reserved
Main Menu 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
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)=x[g(v)]2 and volume (6.2.4) v=xte(y)dy. d g(v)- Figure 6.2.10 Saks,Hille,Etgen Calculus One and Several Vables Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu 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