Page 253 - Calculus Demystified

CHAPTER 8
Applications of the Integral
240
8.5 Arc Length andSurface Area
Just as the integral may be used to calculate planar area and spatial volume, so this
tool may also be used to calculate the arc length of a curve and surface area. The
basic idea is to approximate the length of a curve by the length of its piecewise
linear approximation. A similar comment applies to the surface area. We begin by
describing the basic rubric.

8.5.1 ARC LENGTH

Suppose that f(x) is a function on the interval [a, b]. Let us see how to calculate
the length of the curve consisting of the graph of f over this interval (Fig. 8.28).
We partition the interval:
a = x 0 ≤ x 1 ≤ x 2 ≤ ··· ≤ x k−1 ≤ x k = b.

Look at Fig. 8.29. Corresponding to each pair of points x j−1 ,x j in the partition
is a segment connecting two points on the curve; the segment has endpoints
(x j−1 ,f(x j−1 )) and (x j ,f(x j )). The length j of this segment is given by the

y = f (x)

a b
Fig. 8.28

y = f (x)

(x j , f (x j ))

(x j 1 , f (x j 1 ))
_
_

x x
j –1 j
Fig. 8.29