Copyright © Cengage Learning. All rights reserved. 14 Further Integration Techniques and Applications of the Integral

Copyright © Cengage Learning. All rights reserved Area Between Two Curves and Applications

33 To find the shaded area between the graphs of the two functions as shown in Figure 1, we use the following formula. Area Between Two Graphs If f (x) ≥ g(x) for all x in [a, b] (so that the graph of f does not move below that of g), then the area of the region between the graphs of f and g and between x = a and x = b is given by Figure 1

44 Example 1 – The Area Between Two Curves Find the areas of the following regions: a. Between f (x) = –x 2 – 3x + 4 and g(x) = x 2 – 3x – 4 and between x = –1 and x = 1 b. Between f (x) = | x | and g(x) = –| x – 1| over [–1, 2]

55 Example 1(a) – Solution The area in question is shown in Figure 2. Because the graph of f lies above the graph of g in the interval [–1, 1], we have f (x) ≥ g(x) for all x in [–1, 1]. Therefore, we can use the formula given above and calculate the area as follows: Figure 2

66 Example 1(a) – Solution cont’d

77 Example 1(b) – Solution cont’d The given area can be broken up into triangles and rectangles (see Figure 3), but we already know a formula for the antiderivative of | ax + b | for constants a and b, so we can use calculus instead: Figure 3

88 Example 1(b) – Solution cont’d

99 Area Between Two Curves and Applications Finding the Area Between the Graphs of f (x) and g(x) 1. Find all points of intersection by solving f (x) = g(x) for x. This either determines the interval over which you will integrate or breaks up a given interval into regions between the intersection points. 2. Determine the area of each region you found by integrating the difference of the larger and the smaller function. (If you accidentally take the smaller minus the larger, the integral will give the negative of the area, so just take the absolute value.) 3. Add together the areas you found in step 2 to get the total area.