All Calculus 1 Resources
Example Question #224 : Graphing Functions
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.
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:
By trigonometric identities we get:
Example Question #225 : Graphing Functions
Find the length of the line segment between points A and B:
The distance between two points can be easily found using the Distance Formula:
Applying the points we are given to this formula results in:
This is one of the answer choices.