Arc length of the natural logarithm Consider the curve y = ln x.

a. Find the length of the curve from x =1 to x = a and call it L(a). (Hint: The change of variables allows evaluation by partial fractions.)

b. Graph L(a).

c. As a increases, L(a)increases as what power of a?

Problem 66E

Arc length of the natural logarithm Consider the curve y = ln x.

a. Find the length of the curve from x = 1 to x = a and call it L(a).

b. Graph L(a).

c. As a increases, L(a) increases as what power of a?

Solution

Step 1

a. Find the length of the curve from x = 1 to x = a and call it L(a).

Here we have to find the length of the curve, y = ln x on [1,a]

Consider

Differentiating we get

The length of the curve is given by

Thus … (1)

Take

Differentiating we get,

When we get

When we get