Page 190 - Calculus Workbook For Dummies

174 Part IV: Integration and Infinite Series

*k Use sigma notation to express an 8-right-rectangle approximation of the area under
2
g x = 2 x + 5 from 0 to 4. Then compute the approximation. The notation and approximation
^ h
8
1
are ! i + 20 71.
2
=
4 i 1
=
1. Sketch g xh. You’re on your own.
^
$
2. Express the basic idea of your sum: ! _ base heighti.
8 rectangles
3. Figure the base and plug in.
4 - 0 1
base = =
8 2
1 1
m
!c $ height = ! height
8 2 2 8
4. Express the height as a function of the index of summation, and add the limits of summation:
1 8 1
f ! c im
2 i 1= 2
2
5. Plug in your function, g x = 2 x + 5.
^ h
1 8 1 2
= !> 2 c i + 5H
m
2 i 1 2
=
1 8 1 2 1 8 8 1 2 2 1 1 8 2
2
6. Simplify: = ! c i + ! 5 = !c m i + $ 40 = ! i + 20
m
2 i 1 2 2 i 1 i 1 2 2 4 i 1
=
=
=
=
1 2 8 +
^
1 8 8 + h ^ $ 1h
=
=
7. Use the sum of squares rule to finish: = e o + 20 51 + 20 71
4 6
*l Using your result from problem 11, write a formula for approximating the area under g from
2
188 n + 192 n + 64
0 to 5 with n rectangles. The formula is 2 .
3 n
1. Convert the sigma formula for summing 8 rectangles to one for summing n rectangles.
1 1
Look at Step 5 from the previous solution. The number appears twice. You got when you
2 2
4
computed the width of the base of each rectangle. That’s 4 - 0 , or . You want a formula for
8 8
4 1
n rectangles instead of 8, so use instead of and replace the 8 on top of ! with an n.
n 2
4 n 4 2
m
n !> 2 c n i + 5H
i 1=
4 n 4 2 4 n 4 n 16 2 4 128 n 2
2
2. Simplify: = ! c n im + ! 5 = !d 2 $ 2 i $ n + n 5 $ n = 3 ! i + 20
n
n
n
i 1= i 1= i 1= n n i 1=
3. Use the sum of squares formula.
2
2
^
128 n n + 1 2 ^h n + 1h 128 n + 192 n + 64 188 n + 192 n + 64
= 3 e o + 20 = + 20 =
n 6 3 n 2 3 n 2
m a. Use your result from problem 12 to approximate the area with 50, 100, 1000 and 10,000
rectangles.
2
188 n + 192 n + 64
Area nR =
3 n 2
$
2
$
188 50 + 192 50 + 64
Area 50 = 2
$
R
3 50
. 63 .956