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

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #224 : Graphing Functions

Consider a function  with first-derivative:

.

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

 

Possible Answers:

This is not possible with the information given.

Correct answer:

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:Canvas2

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:

Example Question #225 : Graphing Functions

Find the length of the line segment between points A and B:

 

Possible Answers:

Correct answer:

Explanation:

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.

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