Page 185 - Calculus Workbook For Dummies

169
Chapter 9: Getting into Integration

Q. Estimate the area under f x = lnx from 1 Q. Estimate the area under f x = lnx from 1
^ h
^ h
to 6 with 10 trapezoids. Then compute the to 6 with 10 Simpson rule “trapezoids.”
percent error. Then compute the percent error.
A. The approximate area is 5.733. The error A. The approximate area is 5.751. The error
is about 0.31%. is a mere 0.00069%.
1. Sketch the function and the 1. List the values for a, b, and n, and
10 trapezoids. determine the 21 x-values x 0 through
x 20 , the 11 edges and the 10 base
Already done — Figure 9-1.
midpoints of the 10 curvy-topped
2. List the values for a, b, and n, and “trapezoids.”
determine the 11 x-values, x 0 a = 1
through x 10 (the left edge of the first
trapezoid plus the 10 right edges of b = 6
the 10 trapezoids). n = 20
75
5
25
x 0 = 1 , x 1 = 1 . , x 2 = 1 . , x 3 = 1 . , ... x 20 = 6
Note that in this and all similar problems,
a equals x 0 and b equals x n (x 10 here). 2. Plug these values into the formula.
a = 1 6 - 1
S 20 = 3 20 ( ln1 + 4 ln . 1 25 + 2 ln . 1 5 + 4 ln . 1 75 +
$
b 6 2 ln2 + ... + 4 ln . 5 75 + ln ) 6
=
n = 10 5
. 60 ^ 69 .006202893232 h
5
5
x 0 = 1 , x 1 = 1 . , x 2 = 2 , x 3 = 2 . , ... x 10 = 6
. . 5 7505169
3. Plug these values into the trapezoid
rule formula and solve. 3. Figure the percent error.
The exact answer, again, is
6 - 1
T 10 = (ln1 + 2 ln . 1 5 + 2 ln2 + 2 ln . 2 5 +
$
2 10 5.7505568153635. Round that off to
2 ln3 + 2 ln . 3 5 + 2 ln4 + 2 ln . 4 5 + 5.7505568.
2 ln5 + 2 ln . 5 5 + ln ) 6 . 5 7505568 5 .7505169
-
percent error . .
5 . 5 7505568
. (0 + .811 + . 1 386 + . 1 833 + . 2 197 +
20 .0000069 = .00069 %
. 2 506 + . 2 773 + . 3 008 + . 3 219 +
— way better than either the midpoint or
. 3 409 + . 1 792 ) trapezoid estimate. Impressed?
. . 5 733
4. Compute the percent error.
My TI-89 tells me that the exact area is
5.7505568153635 . . . For this problem,
round that off to 5.751. The percent error
is given by the error divided by the exact
area. So that gives you:
. 5 751 - . 5 733
percent error . . .0031 = . %
31
. 5 751
Compare this to the 10-midpoint-rectangle
error we compute in the solution to
problem 2: 0.14%. In general, the error
with a trapezoid estimate is roughly
twice the corresponding midpoint-
rectangle error.