Page 191 - Calculus Workbook For Dummies

175
Chapter 9: Getting into Integration

Because all right-rectangle estimates with this curve will be over-estimates, this result shows
how far off the approximation of 71 square units was. The answers for the rest of the approxi-
mations are
Area 100 . 63 .309
R
Area 1000 . 62 .731
R
Area ,10 000 . 62 .673
R
b. Now use your result from problem 12 and the definition of the definite integral to determine
the exact area under x2 2 + 5 from 0 to 4. The area is 62.666 . . . or 62 ⁄3.
2
b n
f x dx =
# ^ h lim!= f x i c i b - a mG
_
n
n " 0 i 1
=
a
4 2
2
# _ 2 x + 5i dx = lim 188 n + 192 n + 64
2
n " 3 3 n
0
188
=
3
2
= 62 .6 or 62
3
2
n a. Given the following formulas for left, right, and midpoint rectangles for the area under x + 1
from 0 to 3, approximate the area with 50, 100, 1000, and 10,000 rectangles with each of the
three formulas.
L 50 . 11 .732 R 50 . 12 .272 M 50 . 11 .9991
R
R
R
L 100 . 11 .866 R 100 . 12 .135 M 100 . 11 .999775
R
R
R
L 1000 . 11 .987 R 1000 . 12 .014 M 1000 . 11 .99999775
R
R
R
,000 . 11 .999 ,000 . 12 .001 M 10 000 . 11 .9999999775
L 10 R R 10 R , R
You can see from the results how much better the midpoint-rectangle estimates are than the
other two.
b. Use the definition of the definite integral with each of three formulas from the first part of
1
the problem to determine the exact area under x + from 0 to 3.
2
3 2
2
For left rectangles, # _ x + 1i dx = lim 24 n - 27 n + 9 = 24 = 12
2
n " 3 2 n 2
0
3 2
2
For right rectangles, # _ x + 1i dx = lim 24 n + 27 n + 9 = 24 = 12
2
n " 3 2 n 2
0 3 2
2
And for midpoint rectangles, # _ x + 1i dx = lim 48 n - 9 = 48 = 12
2
n " 3 4 n 4
0
Big surprise — they all equal 12. They better all come out the same since you’re computing
the exact area.
o Continuing with problem 4, estimate the area under y = sinx from 0 to π with eight trapezoids,
and compute the percent error. The approximate area is 1.974 and the error is 1.3%.
1. List the values for a, b, and n, and determine the x-values x 0 through x 8 .
a = 0
b π
=
n = 8
π 2 π 3 π
x 0 = 0 , x 1 = , x 2 = , x 3 = , ... x 8 = π
8 8 8