Chapter 5ET, Slide 1 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.

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Presentation transcript:

Chapter 5ET, Slide 1 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved.

Chapter 5ET, Slide 2 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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.

Chapter 5ET, Slide 3 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.6: The region (a) and solid (b) in Example 4.

Chapter 5ET, Slide 4 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.7: The region (a) and solid (b) in Example 5.

Chapter 5ET, Slide 5 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.8: The region (a) and solid (b) in Example 6.

Chapter 5ET, Slide 6 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.9: The region (a) and solid (b) in Example 7.

Chapter 5ET, Slide 7 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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. baba

Chapter 5ET, Slide 8 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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.

Chapter 5ET, Slide 9 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.12: The inner and outer radii of the washer swept out by the line segment in Figure 5.11.

Chapter 5ET, Slide 10 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.13: The region, limits of integration, and radii in Example 9.

Chapter 5ET, Slide 11 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.14: The washer swept out by the line segment in Figure 5.13.

Chapter 5ET, Slide 12 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.17: Cutting the solid into thin cylindrical slices, working from the inside out. Each slice occurs at some x k between 0 and 3 and has thickness  x. (Example 1)

Chapter 5ET, Slide 13 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. ) 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.

Chapter 5ET, Slide 14 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.19: The shell swept out by the kth rectangle.

Chapter 5ET, Slide 15 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.20: The region, shell dimensions, and interval of integration in Example 2.

Chapter 5ET, Slide 16 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.21: The shell swept out by the line segment in Figure 5.20.

Chapter 5ET, Slide 17 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.22: The region, shell dimensions, and interval of integration in Example 3.

Chapter 5ET, Slide 18 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.23: The shell swept out by the line segment in Figure 5.22.

Chapter 5ET, Slide 19 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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.

Chapter 5ET, Slide 20 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.34: The force F needed to hold a spring under compression increases linearly as the spring is compressed.

Chapter 5ET, Slide 21 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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.

Chapter 5ET, Slide 22 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.37: The olive oil in Example 7.

Chapter 5ET, Slide 23 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.38: (a) Cross section of the glory hole for a dam and (b) the top of the glory hole.

Chapter 5ET, Slide 24 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.39: The glory hole funnel portion.

Chapter 5ET, Slide 25 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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.

Chapter 5ET, Slide 26 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.50: Each mass m, has a moment about each axis.

Chapter 5ET, Slide 27 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.51: A two-dimensional array of masses balances on its center of mass.

Chapter 5ET, Slide 28 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.54: Modeling the plate in Example 3 with vertical strips.

Chapter 5ET, Slide 29 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 5.55: Modeling the plate in Example 3 with horizontal strips.

Chapter 5ET, Slide 30 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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.

Chapter 5ET, Slide 31 Chapter 5 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. 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.