Page 116 - Calculus Demystified

103
The Integral
CHAPTER 4
4.2 Area
Consider the curve shown in Fig. 4.1. The curve is the graph of y = f(x). We set
for ourselves the task of calculating the area A that is (i) under the curve, (ii) above
the x-axis, and (iii) between x = a and x = b. Refer to Fig. 4.2 to see the geometric
region we are considering.

y = f (x)

Fig. 4.1

y = f (x)

a b

Fig. 4.2

We take it for granted that the area of a rectangle of length and width w is
×w. Now our strategy is to divide the base interval [a, b] into equal subintervals.
Fix an integer k> 0. We designate the points

P ={x 0 ,x 1 ,x 2 ,...,x k },
with x 0 = a and x k = b. We require that |x j − x j−1 |=|b − a|/k ≡ x for
j = 1,...,k. In other words, the points x 0 ,x 1 ,...,x k are equally spaced. We call