Page 250 - Calculus Workbook For Dummies

234 Part IV: Integration and Infinite Series

Solutions to Integration Application Problems

x
a What’s the average value of f x = 3 from 1 to 3? The average value is 0.03.
^ h
2
_ x + 1i
3
# x 3
2
total area 1 _ x + 1i
Ave value= =
base 3 - 1
Do this with a u-substitution.
3
1 # 2 x dx
2 1 _ x + 1i 3
2
=
2
10
= # du
1
4 u 3
2
2
=
u x + 1 when x = 1 , u 2 1 10
=
= - u 7 - 2 A
du 2 x dx when x = 3 , u 10 8 2
=
=
1
-
-
2
2
= - _ 10 - 2 i
8
= . 0 03
. 1 7
b A car’s speed in feet per second is given by f t = t - t 6 + 80. What’s its average speed from
^ h
t = 5 seconds to t = 15 seconds? What’s that in miles per hour? . 49 .51 miles per hour.
15
. 1 7
# _ t - t 6 + 80i dt
total distance
Ave speed= = 5
total time 15 - 5
15
1 t - t 3 + 80 t E
2
. 2 7
. 2 7
= 5
10
554 .73 - 675 + 1200 - ^ 28 .57 - 75 + 400h
.
10
. 72 .616 feet per second
. 49 .51 miles per hour
2
1
c What’s the area enclosed by f x = x and g x = x ? The area is ⁄3.
^ h
^ h
1. Graph the functions.
2. Find the points of intersection. I presume you had no trouble finding them: (0, 0) and (1, 1).
3. Find the area. Remember: top minus bottom.
1 1
3
/ 3 2
2
Area= #` x - x j dx = 2 x - 1 x E = 2 - 1 = 1
3 3 3 3 3
0 0
5
3
1
d What’s the total area enclosed by f t = t and g t = t ? The area is ⁄6.
^ h
^ h
1. Graph the functions. You should see three points of intersection.
2. Find the points: (–1, –1), (0, 0), and (1, 1).
3. Find the area on the left.
5
t is on top, so
0 0
3
6
4
5
Area= # _ t - t i dt = 1 t - 1 t E = 0 - c 1 - 1 m = 1
6 4 6 4 12
- 1 - 1