# Calculus 1 : How to find length of line by graphing functions

## Example Questions

### Example Question #1 : Length Of Line

Consider a function  with first-derivative:

.

Which integral can calculate the length along this curve between  and ?

This is not possible with the information given.

Explanation:

To determine the length of a curve between two points, we evaluate the integral

There are many reasons this works, but we'll give an informal explanation here:

If we divide this curve into three line-segments, we can see that they become more and more similar to the original curve. By adding up all the little hypotenuses, we can arrive at the length of the curve. Note, that

If we think of the integration symbol as a sum of infinitely small parts, this gets us the formula for length:

.

Returning to the problem, we plug the derivative into the length formula:

substitute:

Simplify:

By trigonometric identities we get: