Page 251 - Calculus Demystified

CHAPTER 8
Applications of the Integral
238 This last is a Riemann sum for the integral (1/[b − a]) · a b f(x) dx. Thus, letting
the mesh of the partition go to zero, we declare
1 b
average of f = σ = b − a f(x) dx.
a
EXAMPLE 8.17
In a tropical rain forest, the rainfall at time t isgiven by ϕ(t) = 0.1 − 0.1t +
2
0.05t inchesper hour, 0 ≤ t ≤ 10. What isthe average rainfall for times
0 ≤ t ≤ 6?
SOLUTION
We need only average the function ϕ:
6
average rainfall = σ = 1 ϕ(t) dt
6 − 0 0
= 1 6 0.1 − 0.1t + 0.05t dt
2
6 0
1 2 0.05 3 6
= 6 0.1t − 0.05t + 3 t 0
TEAMFLY
= 0.1 − 0.3 + 0.6

= 0.4 inches per hour.
EXAMPLE 8.18
Let f(x) = x/2 − sin x on the interval [−2, 5]. Compare the average value
of thisfunction on the interval with itsminimum and maximum.
SOLUTION
Observe that
1

f (x) = 2 − cos x.
Thus the critical points occur when cos x = 1/2, or at −π/3,π/3. We also
must consider the endpoints −2, 5. The values at these points are

f(−2) =−1 + sin 2 ≈−0.0907026
√
π 3
f(−π/3) =− 6 + √ 2 ≈ 0.3424266
π 3
f(π/3) = 6 − 2 ≈−0.3424266
5
f(5) = 2 − sin 5 ≈ 3.458924.

Team-Fly
®