Chapter 5, Slide 1 Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001....
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Transcript of Chapter 5, Slide 1 Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001....
Chapter 5, Slide 1Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Finney Weir GiordanoFinney Weir Giordano
Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 5, Slide 2Chapter 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.
Chapter 5, Slide 3Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.6: The region (a) and solid (b) in Example 4.
Chapter 5, Slide 4Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.7: The region (a) and solid (b) in Example 5.
Chapter 5, Slide 5Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.8: The region (a) and solid (b) in Example 6.
Chapter 5, Slide 6Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.9: The region (a) and solid (b) in Example 7.
Chapter 5, Slide 7Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. 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. b
a
Chapter 5, Slide 8Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. 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 5, Slide 9Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. 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 5, Slide 10Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.13: The region, limits of integration, and radii in Example 9.
Chapter 5, Slide 11Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.14: The washer swept out by the line segment in Figure 5.13.
Chapter 5, Slide 12Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. 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 xk between 0 and 3 and has thickness x. (Example 1)
Chapter 5, Slide 13Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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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 5, Slide 14Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.19: The shell swept out by the kth rectangle.
Chapter 5, Slide 15Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.20: The region, shell dimensions, and interval of integration in Example 2.
Chapter 5, Slide 16Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.21: The shell swept out by the line segment in Figure 5.20.
Chapter 5, Slide 17Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.22: The region, shell dimensions, and interval of integration in Example 3.
Chapter 5, Slide 18Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.23: The shell swept out by the line segment in Figure 5.22.
Chapter 5, Slide 19Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.32: The force F needed to hold a spring under compression increases linearly as the spring is compressed.
Chapter 5, Slide 20Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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.
Chapter 5, Slide 21Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.35: The olive oil in Example 7.
Chapter 5, Slide 22Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.36: (a) Cross section of the glory hole for a dam and (b) the top of the glory hole.
Chapter 5, Slide 23Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.37: The glory hole funnel portion.
Chapter 5, Slide 24Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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.
Chapter 5, Slide 25Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.48: Each mass m, has a moment about each axis.
Chapter 5, Slide 26Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.49: A two-dimensional array of masses balances on its center of mass.
Chapter 5, Slide 27Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.52: Modeling the plate in Example 3 with vertical strips.
Chapter 5, Slide 28Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 5.53: Modeling the plate in Example 3 with horizontal strips.
Chapter 5, Slide 29Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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.
Chapter 5, Slide 30Chapter 5. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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.