1. Copyright © Cengage Learning. All rights reserved.
Integration
Copyright © Cengage Learning. All rights reserved.

2. Numerical Integration
Copyright © Cengage Learning. All rights reserved.

3. Objectives
Mr. Z says “don’t use Trapezoidal rule on AP test, but here it is.”
Approximate a definite integral using the Trapezoidal Rule.

4. Mr. Z and Trapezoids
Work Example 1 in textbook (page 306) with regular trapezoids

5. The Trapezoidal Rule

6. The Trapezoidal Rule
One way to approximate a definite integral is to use n trapezoids, as shown in Figure 4.42.
In the development of this method,
assume that f is continuous and
positive on the interval [a, b].
So, the definite integral
represents the area of the region
bounded by the graph of f and the
x-axis, from x = a to x = b.
Figure 4.42

7. The Trapezoidal Rule
First, partition the interval [a, b] into n subintervals, each of width ∆x = (b – a)/n, such that
Then form a trapezoid for each
subinterval (see Figure 4.43).
Figure 4.43

8. The Trapezoidal Rule The area of the ith trapezoid is
This implies that the sum of the areas of the n trapezoids

9. The Trapezoidal Rule Letting you can take the limits as to obtain
The result is summarized in the following theorem.

10. The Trapezoidal Rule

11. Example 1 – Approximation with the Trapezoidal Rule
Use the Trapezoidal Rule to approximate
Compare the results for n = 4 and n = 8, as shown in Figure 4.44.
Figure 4.44

12. Example 1 – Solution
When n = 4, ∆x = π/4, and you obtain

13. Example 1 – Solution
cont’d
When and you obtain

14. So why do it with a table? AP test tries to trick us
Example with a table provided by Mr. Z

15. Homework 54
Homework: page 310: 1, 5, 7, 9

16. Homework 55
Homework: page AP4-1: 1-10