Applications of Integration CHAPTER 6

2 Copyright © Houghton Mifflin Company. All rights reserved. 6.1 Area of a Region Between Two Curves Objectives:  Find the area of a region between two curves using integration.  Find the area of a region between intersecting curves using integration.  Describe integration as an accumulation process.

3 Copyright © Houghton Mifflin Company. All rights reserved. Area Under a Curve  Write the expression to find the area under the curve f (x) from a to b.  Area under a curve:

4 Copyright © Houghton Mifflin Company. All rights reserved. Find the Area Between Two Curves

5 Copyright © Houghton Mifflin Company. All rights reserved. Graphical Representation

6 Copyright © Houghton Mifflin Company. All rights reserved. Area of a Region Between Two Curves Upper curve – Lower curve Regardless of the relative position of the x -axis.

7 Copyright © Houghton Mifflin Company. All rights reserved. The Variations 1.Functions f (x) and g(x) do not intersect. Values a and b are given explicitly. 2.Functions f (x) and g(x) intersect. Values a and b must be calculated algebraically or using INTERSECT feature of graphing calculator. 3.Functions f (x) and g(x) intersect at more than two points. All points of intersection must be calculated. Must identify which curve is above the other in each interval.

8 Copyright © Houghton Mifflin Company. All rights reserved. Example 1  Sketch the region bounded by the graphs of the functions and find the area of the region.

9 Copyright © Houghton Mifflin Company. All rights reserved. Example 2  Sketch the region bounded by the graphs of the functions and find the area of the region. Find intersection points algebraically.

10 Copyright © Houghton Mifflin Company. All rights reserved. Example 3  Sketch the region bounded by the graphs of the functions and find the area of the region. Find intersection points with the calculator.

11 Copyright © Houghton Mifflin Company. All rights reserved. Example 4  Sketch the region bounded by the graphs of the functions and find the area of the region.  When working with functions of y, integrate with respect to y, subtracting the right curve minus the left curve.

12 Copyright © Houghton Mifflin Company. All rights reserved. The Accumulation Function F (x)  The accumulation function of a function f (t), gives the accumulation of the area between the horizontal axis and the graph of f from a to x. (“Area accumulated so far.”)  The constant a is referred to as the starting value of the accumulation.

13 Copyright © Houghton Mifflin Company. All rights reserved. Animation  Link to animation:

14 Copyright © Houghton Mifflin Company. All rights reserved. Applications of the Accumulation Function  If f (x) represents the rate of consumption of a beverage… o Then the integral of f is the actual amount of the beverage consumed in the time interval.  If f (x) represents a car’s velocity during a given time interval… o Then the integral of f is the total distance traveled during that time.

15 Copyright © Houghton Mifflin Company. All rights reserved. Example  Find the accumulation function F. Then evaluate F at each specified value of the independent variable and graphically show the area given by each value of F. a)F (0) b)F (4) c)F (6)

16 Copyright © Houghton Mifflin Company. All rights reserved. Homework Pp. 418 – 419 # 15 – 49 odd # 53, 55, 56