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Chapter 4, Slide 1Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Finney Weir GiordanoFinney Weir Giordano
Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 4, Slide 2Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.1: The curves y = x3 + C fill the coordinate plane without overlapping. In Example 4, we identify the curve y = x3 – 2 as the one that passes through the given point (1, –1).
Chapter 4, Slide 3Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.3: The region under the concentration curve of Figure 4.2 is approximated with rectangles. We ignore the portion from t = 29 to t = 31; its concentration is negligible.
Chapter 4, Slide 4Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.4: (a) The semicircle y = 16 – x2 revolved about the
x-axis to outline a sphere. (b) The solid sphere approximated with cross-section based cylinders.
Chapter 4, Slide 5Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.5: (a) The graph of ƒ(x) = x2, –1 x 1. (b) Values of ƒ sampled at regular intervals.
Chapter 4, Slide 6Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.7: The graph of a typical function y = ƒ(x) over a closed interval [a, b]. The rectangles approximate the region between the graph of the function and the x-axis.
Chapter 4, Slide 7Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.8: The curve of Figure 4.7 with rectangles from finer partitions of [a, b]. Finer partitions create more rectangles with shorter bases.
Chapter 4, Slide 8Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.11: A sample of values of a function on an interval [a, b].
Chapter 4, Slide 9Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.16: The rate at which the wiper blade on a bus clears the windshield of rain as the blade moves past x is the height of the blade. In symbols, dA/dx = ƒ(x).
Chapter 4, Slide 10Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.18: The graph of the household voltage V = Vmax sin 120 t over a full cycle. Its average value over a half-cycle is 2Vmax/. Its average value over a full cycle is zero.
Chapter 4, Slide 11Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.21: Ak = area of k th rectangle,ƒ(ck) – g(ck ) = height, xk = width.
Chapter 4, Slide 12Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.23: When the formula for a bounding curve changes, the area integral changes to match. (Example 5)
Chapter 4, Slide 13Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.24: The Trapezoidal Rule approximates short stretches of the curve y = ƒ(x) with line segments. To approximate the integral of ƒ from a to b, we add the “signed” areas of the trapezoids made by joining the ends of the segments to the x-axis.
Chapter 4, Slide 14Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.27: Simpson’s Rule approximates short stretches of curve with parabolic arcs.
Chapter 4, Slide 15Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.28: By integrating from –h to h, we find the shaded area to be
( y0 + 4y1 + y2).h3
Chapter 4, Slide 16Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 4.32: Rolling and unrolling a carpet: a geometric interpretation of Leibniz’s Rule.
