Chapter 5 ET. 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 5.31: Slope fields (top row) and selected solution curves (bottom row). In computer renditions, slope segments are sometimes portrayed with vectors, as they are here. This is not to be taken as an indication that slopes have directions, however, for they do not.

Figure 5.34: The force F needed to hold a spring under compression increases linearly as the spring is compressed.

Figure 5.36: 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 5.37: The olive oil in Example 7.

Figure 5.38: (a) Cross section of the glory hole for a dam and (b) the top of the glory hole.

Figure 5.39: The glory hole funnel portion.

Figure 5.45: 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 5.50: Each mass m, has a moment about each axis.

Figure 5.51: A two-dimensional array of masses balances on its center of mass.

Figure 5.54: Modeling the plate in Example 3 with vertical strips.

Figure 5.55: Modeling the plate in Example 3 with horizontal strips.

Figure 5.56: 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. 57: The semicircular wire in Example 6 Figure 5.57: 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.