Suppose you are asked to find the length of the curve y = f (x) from x = a to x = b. To do this, we could divide the interval [a, b] into a finite number of subintervals. To each endpoint of each subinterval there is a corresponding y-value. If we draw line segments from one y-value to the next, we will create a polygonal path. Figure 1 illustrates this.

Each little line segment of this path has a length. If I sum up all of these lengths, I will get an approximate length of the curve. As the partition becomes finer, the total length of this curve approaches the true length of the curve. In fact the

figure 1

length of the curve can be determined by the following integral for y = f (x) on the interval [a, b].

EXAMPLE 1:

Find the length of the curve

from x = 0 to x = 3.

SOLUTION:

EXAMPLE 2:

Find the length of the curve

from x = 1 to x = 8.

SOLUTION:

 What if dy/dx fails to exist at a point on the curve? Well, dx/dy might exist at that point. If it does, then solve for x in terms of y and then use the following formula for x = g (y), c £ y £ d.

EXAMPLE 3:

Find the length of the curve

from y = 1 to y = 3.

SOLUTION:

EXAMPLE 4:

Find the length of the curve

from y = -p /4 to y = p /4.

SOLUTION:

(This comes from the Fundamental Theorem of Calculus, Part 1.)

1 + (g' (y)) ² = 1 + sec ⁴ y - 1 = sec ⁴ y