1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 6
The Definite Integral
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter Outline
Antidifferentiation
The Definite Integral and Net Change of a Function
The Definite Integral and Area under a Graph
Areas in the xy-Plane
Applications of the Definite Integral
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 6.4
Areas in the xy-Plane
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section Outline
Properties of Definite Integrals
Area Between Two Curves
Finding the Area Between Two Curves
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Definite Integral
Geometric interpretation of the definite integral:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Area Between Two Curves
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Finding the Area Between Two Curves
EXAMPLE
Find the area of the region between y = 2x2 – 4x+6 and y = – x2 + 2x + 1 from x = 1 to x = 2.
SOLUTION
Upon sketching the two graphs (Fig. 7), we see that f(x) = 2x2 − 4x + 6 lies above g(x) = −x2 +2x+1 for
1 ≤ x ≤ 2. Therefore, our formula gives the area of the shaded region as
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Finding the Area Between Two Curves
EXAMPLE
Find the area of the region between y = 2x2 – 4x+6 and y = – x2 + 2x + 1 from x = 1 to x = 2.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Finding the Area Between Two Curves
EXAMPLE
Write down a definite integral or sum of definite integrals that gives the area of the shaded portion of the figure.
SOLUTION
Since the two shaded regions are (1) disjoint and (2) have different functions on top, we will need a separate integral for each. Therefore
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Finding the Area Between Two Curves
CONTINUED
Therefore, to represent all the shaded regions, we have
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Finding the Area Between Two Curves
EXAMPLE
Two rockets are fired simultaneously straight up into the air. Their velocities (in meters per second) are v1(t) and v2(t), respectively, and v1(t) ≥ v2(t) for t ≥ 0. Let A denote the area of the region between the graphs of y = v1(t) and y = v2(t) for 0 ≤ t ≤ 10. What physical interpretation may be given to the value of A?
SOLUTION
Since v1(t) ≥ v2(t) for t ≥ 0, this suggests that the first rocket is always traveling at least as fast as the second rocket. Therefore, we have
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Finding the Area Between Two Curves
CONTINUED
But again, since v1(t) ≥ v2(t) for t ≥ 0, we know that So,
this implies that This means that the position of the first rocket is always at least as high (up in the air) as that of the second rocket. That is, the first rocket is always higher up than the second rocket (or at the same height).
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

13. Review 0.4 Zeros of Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

14. Review 0.4 Zeros of Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

15. Review 0.4 Zeros of Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.