1. Chapter 4, Slide 1 Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Finney Weir Giordano Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.

2. Chapter 4, Slide 2 Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 4.1: The curves y = x 3 + C fill the coordinate plane without overlapping. In Example 4, we identify the curve y = x 3 – 2 as the one that passes through the given point (1, –1).

3. Chapter 4, Slide 3 Chapter 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.

4. Chapter 4, Slide 4 Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 4.4: (a) The semicircle y =  16 – x 2 revolved about the x-axis to outline a sphere. (b) The solid sphere approximated with cross-section based cylinders.

5. Chapter 4, Slide 5 Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 4.5: (a) The graph of ƒ(x) = x 2, –1  x  1. (b) Values of ƒ sampled at regular intervals.

6. Chapter 4, Slide 6 Chapter 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.

7. Chapter 4, Slide 7 Chapter 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.

8. Chapter 4, Slide 8 Chapter 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].

9. Chapter 4, Slide 9 Chapter 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).

10. Chapter 4, Slide 10 Chapter 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 = V max sin 120  t over a full cycle. Its average value over a half-cycle is 2V max / . Its average value over a full cycle is zero.

11. Chapter 4, Slide 11 Chapter 4. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 4.21:  A k = area of k th rectangle,  ƒ(c k ) – g(c k ) = height,  x k = width.

12. Chapter 4, Slide 12 Chapter 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)

13. Chapter 4, Slide 13 Chapter 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.

14. Chapter 4, Slide 14 Chapter 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.

15. Chapter 4, Slide 15 Chapter 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 ( y 0 + 4y 1 + y 2 ). h3h3

16. Chapter 4, Slide 16 Chapter 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.