Chapter 4 ET. 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).

Figure 4. 3: The region under the concentration curve of Figure 4 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.

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.

Figure 4. 5: (a) The graph of ƒ(x) = x2, –1 x  1 Figure 4.5: (a) The graph of ƒ(x) = x2, –1 x  1. (b) Values of ƒ sampled at regular intervals.

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.

Figure 4. 8: The curve of Figure 4 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.

Figure 4.11: A sample of values of a function on an interval [a, b].

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).

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.

Figure 4.21: Ak = area of kth rectangle, ƒ(ck) – g(ck ) = height, xk = width.

Figure 4.23: When the formula for a bounding curve changes, the area integral changes to match. (Example 5)

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.

Figure 4.27: Simpson’s Rule approximates short stretches of curve with parabolic arcs.

Figure 4.28: By integrating from –h to h, we find the shaded area to be ( y0 + 4y1 + y2). h 3

Figure 4.32: Rolling and unrolling a carpet: a geometric interpretation of Leibniz’s Rule.