Page 119 - Calculus Demystified

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Thus we have learned that CHAPTER 4 The Integral

b
area A = f(x) dx.
a
It is well to take a moment and comment on the integral notation. First, the
integral sign

is an elongated “S,” coming from “summation.” The dx is an historical artifact,
coming partly from traditional methods of developing the integral, and partly from
a need to know explicitly what the variable is. The numbers a and b are called the
limits of integration—the number a is the lower limit and b is the upper limit. The
function f is called the integrand.
Before we can present a detailed example, we need to record some important
information about sums:
N
I. We need to calculate the sum S = 1 + 2 + ··· + N = j=1 j. To achieve this
goal, we write

S = 1 + 2 + ··· + (N − 1) + N
S = N + (N − 1) + ··· + 2 + 1
Adding each column, we obtain

2S = (N + 1) + (N + 1) + ··· + (N + 1) + (N + 1) .

N times
Thus
2S = N · (N + 1)

or
N · (N + 1)
S = .
2
This is a famous formula that was discovered by Carl Friedrich Gauss (1777–1855)
when he was a child.
2 2 2 n 2
II. The sum S = 1 + 2 + ··· + N = j=1 j is given by
2
3
2N + 3N + N
S = .
6
We shall not provide the details of the proof of this formula, but refer the interested
reader to [SCH2].