Page 236 - Calculus Workbook For Dummies

220 Part IV: Integration and Infinite Series

1. What’s the average value of f x = x 3 2. A car’s speed in feet per second is given
^ h
2
. 1 7
from 1 to 3? _ x + 1i by f t = t - t 6 + 80. What’s its average
^ h
speed from t = 5 seconds to t = 15 seconds?
Solve It What’s that in miles per hour?
Solve It

Finding the Area between Curves

In elementary school and high school geometry, you learned area formulas for all sorts
of shapes like rectangles, circles, triangles, parallelograms, kites, and so on. Big deal.
2
With integration, you can determine things like the area between f x = x and
^ h
g x = arctanx — now that is something.
^ h
Q. What’s the area between sinx and cosx / π 4
h
Area= #^ cosx - sinx dx
from 0 to π?
0
/ π 4
A. The area is 2 2. = sinx + cosx@ 0
2 2
1. Graph the two functions to get a feel for = + - ^ 0 + 1h
the size of the area in question and 2 2
where the functions intersect. = 2 - 1
π
2. Find the point of intersection. 4. Figure the area between and π.
4
sinx = cosx This time sine’s on top:
π
sinx
cosx = 1 Area= #^ sinx - cosx dx
h
tanx = 1 / π 4
π
π = - cosx - sinx@
x = / π 4
4 J N
1 - - -
=- - h 0 K K 2 - 2 O
^
π
3. Figure the area from 0 to . L 2 2 O
P
π 4
Between 0 and , cosine is on top so you = 1 + 2
4
want cosine minus sine:
5. Add the two areas for your final answer.
1
1
2 - + + 2 = 2 2