Page 252 - Calculus Demystified

CHAPTER 8
Applications of the Integral
Plainly, the maximum value is f(5) = 5/2−sin 5 ≈ 3.458924. The minimum 239
value is f(π/3) ≈−0.3424266.
The average value of our function is
1 5 x
σ = − sin xdx
5 − (−2) −2 2

2 5
1 x
= + cos x
7 4 −2

1 25 4
= + cos 5 − + cos 2
7 4 4

1 21
= + cos 5 − cos 2
7 4
≈ 0.84997.

You can see that the average value lies between the maximum and the minimum,
as it should. This is an instance of a general phenomenon.

You Try It: On a certain tree line, the height of trees at position x is about 100 −
3x + sin 5x. What is the average height of trees from x = 2to x = 200?

EXAMPLE 8.19
What isthe average value of the function g(x) = sin x over the interval
[0, 2π]?

SOLUTION
We calculate that

2π 2π
1 1 1
σ = sin xdx = [− cos x] = [−1 − (−1)]= 0.
2π − 0 0 2π 0 2π
We see that this answer is consistent with our intuition: the function g(x) =
sin x takes positive values and negative values with equal weight over the
interval [0, 2π]. The average is intuitively equal to zero. And that is the actual
computed value.

You Try It: Give an example of a function on the real line whose average over
every interval of length 4 is 0.