Page 183 - Calculus Workbook For Dummies

167
Chapter 9: Getting into Integration

Q. The answer for the example in the last calculus — actually (sort of) adding up
section gives the approximate area under an infinite number of rectangles.
2
f x = x + 3 x from 0 to 5 given by n b n b - a
^ h
f x dx =
_
i
2
475 n + 600 n + 125 # ^ h lim!= f x i $ c n mG
rectangles as 2 . For a n " 3 i 1
=
6 n
5 2
20 rectangles, you found the approximate # _ x + 3 i limd 475 n + 600 n + 125 n
2
x dx =
area of ~84.2. With this formula and your n " 3 6 n 2
0
calculator, compute the approximate area 475
given by 50, 100, 1000, and 10,000 rectan- = 6
gles, then use the definition of the definite 1
integral to compute the exact area. = 79 .16 or 79 6
A. The exact area is 79 .16 . The answer of 475 follows immediately
6
from the horizontal asymptote rule (see
$
2
$
475 50 + 600 50 + 125
Area 50 = 2 Chapter 4). You can also break the frac-
R
6 50
$
= 81 .175 tion in line two above into three pieces
and do the limit the long way:
Area 100 . 80 .169
R
2
Area 1000 . 79 .267 = limd 475 n 2 + 600 n 125 2 n
2 +
R
x " 3 6 n 6 n 6 n
Area 10000 . 79 .177
R
= lim 475 + lim 100 + lim 125 2
n
These estimates are getting better and x " 3 6 x " 3 x " 3 6 n
better; they appear to be headed toward = 475 + 0 + 0
something near 79. Now for the magic of 6
475
=
6
13. In problem 11, you estimate the area under 14. a. Given the following formulas for n left,
5
g x = 2 x + 5 from 0 to 4 with 8 rectangles. right, and midpoint rectangles for the
^ h
The result is 71 square units. area under x + from 0 to 3, approxi-
2
1
mate the area with 50, 100, 1000, and
a. Use your result from problem 12 to 10,000 rectangles with each of the three
approximate the area under g with 50, formulas:
100, 1000, and 10,000 rectangles.
2
24 n - 27 n + 9
b. Now use your result from problem 12 L nR = 2 n 2
and the definition of the definite integral 24 n + 27 n + 9
2
to determine the exact area under R nR = 2 n 2
2
2 x + 5 from 0 to 4. 2
M nR = 48 n - 9
2
Solve It 4 n
b. Use the definition of the definite integral
with each of three formulas from the
first part of the problem to determine
1
2
the exact area under x + from 0 to 3.
Solve It