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. Definite Integral and Area under a Graph
Section 6.3
Definite Integral and Area under a Graph
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section Outline
Area Under a Graph
Calculating Definite Integrals
The Fundamental Theorem of Calculus
Area Under a Curve as an Antiderivative
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Area Under a Graph
Definition
Example
Area Under the Graph of f (x) from a to b: An example of this is shown to the right
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Theorem I: Area under a Graph
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Area Under the Graph of a Function
EXAMPLE
Use Theorem 1 to compute the area of the shaded region under the graph of
f(x) = 3x2 + ex, from x = −1 to x = 1.
SOLUTION
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Riemann Sums
In this section we will learn to estimate the area under the graph of f (x) from x = a to x = b by dividing up the interval into partitions (or subintervals),
each one having width where n = the number of partitions that
will be constructed. In the example below, n = 4.
A Riemann Sum is the sum of the areas of the rectangles generated above.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Riemann Sums to Approximate Areas
EXAMPLE
Use a Riemann sum to approximate the area under the graph f (x) on the given interval using midpoints of the subintervals
SOLUTION
The partition of -2 ≤ x ≤ 2 with n = 4 is shown below. The length of each subinterval is x1 x2 x3 x4 -2 2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Riemann Sums to Approximate Areas
CONTINUED
Observe the first midpoint is units from the left endpoint, and the midpoints themselves are units apart. The first midpoint is x1 = = = Subsequent midpoints are found by successively adding
midpoints: -1.5, -0.5, 0.5, 1.5
The corresponding estimate for the area under the graph of f (x) is
So, we estimate the area to be 5 (square units).
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Approximating Area With Midpoints of Intervals
CONTINUED
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Riemann Sums to Approximate Areas
EXAMPLE
Use a Riemann sum to approximate the area under the graph f (x) on the given interval using left endpoints of the subintervals
SOLUTION
The partition of 1 ≤ x ≤ 3 with n = 5 is shown below. The length of each subinterval is 1
1.4
1.8
2.2
2.6 3 x1 x2 x3 x4 x5
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Riemann Sums to Approximate Areas
CONTINUED
The corresponding Riemann sum is
So, we estimate the area to be (square units).
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Approximating Area Using Left Endpoints
CONTINUED
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Fundamental Theorem of Calculus
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Fundamental Theorem of Calculus
EXAMPLE
Use the Fundamental Theorem of Calculus to calculate the following integral.
SOLUTION
An antiderivative of 3x1/3 – 1 – e0.5x is Therefore, by the fundamental theorem,
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Fundamental Theorem of Calculus
EXAMPLE
(Heat Diffusion) Some food is placed in a freezer. After t hours the temperature of the food is dropping at the rate of r(t) degrees Fahrenheit per hour, where
(a) Compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2.
(b) What does the area in part (a) represent?
SOLUTION
(a) To compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2, we evaluate the following.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Fundamental Theorem of Calculus
CONTINUED
(b) Since the area under a graph can represent the amount of change in a quantity, the area in part (a) represents the amount of change in the temperature between hour t = 0 and hour t = 2. That change is degrees Fahrenheit.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Fundamental Theorem of Calculus
Copyright © 2014, 2010, 2007 Pearson Education, Inc.