Clicker Question 1

What is
–
–
–
–
–
 x sec( x
2
)dx
?
A. x tan(x2) + C
B. ½ (sec(x2) + tan(x2)) + C
C. ln |sec(x2) + tan(x2)| + C
D. ½ tan(x2) + C
E. ½ ln |sec(x2) + tan(x2)| + C
Clicker Question 2

What is
–
–
–
–
–
2
?
x
sec
(
x
)
dx

A. x2 tan(x) + C
B. x tan(x) + sec2(x) + C
C. x tan(x) – ln |sec(x)| + C
D. ½ x2 tan(x) + C
E. x tan(x) + ln |sec(x)| + C
Numerical (or Approximate) Integration
(2/18/11)


To evaluate a definite integral, we always
hope to apply the Fundamental Theorem by
finding an antiderivative and then evaluating
it at the endpoints. But this isn’t always
possible or practical.
Another option always available is to do
numerical integration to approximate the
answer. Here we use Riemann sums.
Getting Good Approximations


In general, the more data you use (i.e., the
more rectangles you measure), the better the
estimate.
Methods (in increasing order of accuracy for
a fixed number of rectangles)
–
–
–
–
Left or right-hand endpoints
Trapezoid Rule (average of the above)
Midpoint Rule
Simpson’s Rule
Simpson’s Rule




It turns out that the Midpoint Rule is about
twice as accurate as the Trapezoid Rule and
the errors are in opposite directions.
Hence we form a “weighted average”:
Simpson’s Rule = (2 Midpoint + Trapezoid) / 3
Simpson’s Rule is in general extremely
accurate even for small n (i.e., few rectangles).
Can get Simpson directly from the data using
(1 + 4 + 2 + 4 + 2 +…+ 2 + 4 + 1) / 6
Clicker Question 3

Suppose a definite integral has the following
estimates for 10 subdivisions of the interval:
Left-hand sum: 12.5
Right-hand sum: 16.5
Midpoint Rule sum: 13.0
What is the Simpson’s Rule estimate?

A. 13.5 B. 13.0 C. 13.75 D. 14.0 E. 13.25
Assignment for Monday



Work on Hand-in #2 (due next Tuesday at 4:30)
Section 7.7 goes into much more detail on
approximate integration than we need to, and
defines Simpson’s Rule slightly differently, so just
work from our notes please.
Do Exercise 3 on page 505 (note: no known way to
use FTC on this one), but also compute the
Simpson’s Rule estimate with n = 4 both using the
weighted average of Trap and Mid and the
(1-4-2-4-2-4-2-4-1) / 6 method.