Finney Weir Giordano Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.

Figure 5.5: The wedge of Example 3, sliced perpendicular to the x-axis. The cross sections are rectangles.

Figure 5.6: The region (a) and solid (b) in Example 4.

Figure 5.7: The region (a) and solid (b) in Example 5.

Figure 5.8: The region (a) and solid (b) in Example 6.

Figure 5.9: The region (a) and solid (b) in Example 7.

Figure 5.10: The cross sections of the solid of revolution generated here are washers, not disks, so the integral A(x) dx leads to a slightly different formula. b a

Figure 5.11: The region in Example 8 spanned by a line segment perpendicular to the axis of revolution. When the region is revolved about the x-axis, the line segment will generate a washer.

Figure 5.12: The inner and outer radii of the washer swept out by the line segment in Figure 5.11.

Figure 5.13: The region, limits of integration, and radii in Example 9.

Figure 5.14: The washer swept out by the line segment in Figure 5.13.

Figure 5.17: Cutting the solid into thin cylindrical slices, working from the inside out. Each slice occurs at some xk between 0 and 3 and has thickness  x. (Example 1)

Figure 5.18: Imagine cutting and unrolling a cylindrical shell to get a (nearly) flat rectangular solid. Its volume is approximately v = length  height  thickness. )

Figure 5.19: The shell swept out by the kth rectangle.

Figure 5.20: The region, shell dimensions, and interval of integration in Example 2.

Figure 5.21: The shell swept out by the line segment in Figure 5.20.

Figure 5.22: The region, shell dimensions, and interval of integration in Example 3.

Figure 5.23: The shell swept out by the line segment in Figure 5.22.

Figure 5.32: The force F needed to hold a spring under compression increases linearly as the spring is compressed.

Figure 5.34: To find the work it takes to pump the water from a tank, think of lifting the water one thin slab at a time.

Figure 5.35: The olive oil in Example 7.

Figure 5.36: (a) Cross section of the glory hole for a dam and (b) the top of the glory hole.

Figure 5.37: The glory hole funnel portion.

Figure 5.43: To find the force on one side of the submerged plate in Example 2, we can use a coordinate system like the one here.

Figure 5.48: Each mass m, has a moment about each axis.

Figure 5.49: A two-dimensional array of masses balances on its center of mass.

Figure 5.52: Modeling the plate in Example 3 with vertical strips.

Figure 5.53: Modeling the plate in Example 3 with horizontal strips.

Figure 5.54: Modeling the plate in Example 4 with (a) horizontal strips leads to an inconvenient integration, so we model with (b) vertical strips instead.

Figure 5. 55: The semicircular wire in Example 6 Figure 5.55: The semicircular wire in Example 6. (a) The dimensions and variables used in finding the center of mass. (b) The center of mass does not lie on the wire.