Page 250 - Calculus Demystified

Applications of the Integral
CHAPTER 8
We calculate the total work by adding all these elements together using an 237
integral. The result is
3

2
W = 62.4π · (100 − x ) · x dx
1

3
3
= 62.4π · 100x − x dx
1
3
4
x
2
= 62.4π 50x −
4 1

81 1
= 62.4π 450 − − 50 −
4 4
= 23,712π foot-pounds.
You Try It: A spring has Hooke’s constant 5. How much work is performed in
stretching the spring half a foot from its rest position?

8.4 Averages

In ordinary conversation, when we average a collection p 1 ,...,p k of k numbers,
we add them together and divide by the number of items:
p 1 + ··· + p k
σ = Average = .
k
The significance of the number σ is that if we wanted all the k numbers to be equal,
but for the total to be the same, then that common value would have to be σ.
Now suppose that we want to average a continuous function f over an interval
[a, b] of its domain. We can partition the interval,
P ={x 0 ,x 1 ,...,x k },

with x 0 = a and x k = b as usual. We assume that this is a uniform partition, with
x j −x j−1 = x = (b−a)/k for all j. Then an “approximate average” of f would
be given by
f(x 1 ) + f(x 2 ) + ··· + f(x k )
σ app = .
k
It is convenient to write this expression as
k k
1 b − a 1
σ app = f(x j ) · = f(x j ) · 8x.
b − a k b − a
j=1 j=1