Arc length and definite integral

[sithe]

Hi everyone ... I am a first time poster from South Africa. I have been visiting the forum for some time. I am busy teaching myself calculus and physics.

I have hiccup with the concept of definite integral and arc length of a function. In my understanding these should be the same thing.

But looking at the formulas for calculating each, they appear to be different things. What am I missing?

Are they the same thing or different things?

Thanks,
Sithe

 

[sweet springs]

Hi.

Not arc length but area, isn't it?

Regards.

 

[micromass]

The definite integral is supposed to be the area. One can use it to calculate arc length, but that's not straightforward.

Can you tell us why the two should be the same thing?

 

[JJacquelin]

This is shown on the joint figure :

 

[sithe]

Thank you JJacquelin ... that makes it perfectly clear now.

micromass ... the book I read made an example as follows:

A car travels at constant speed of 100 miles/hour, after two hours the car has traveled 200 miles. I paraphrase ... "it turns out that the area under the curve (where curve is drawn on time / speed axis) is equal to the length of the curve."

I guess the above claim only holds for a constant curve.

 

[chiro]

Thank you JJacquelin ... that makes it perfectly clear now.

micromass ... the book I read made an example as follows:

A car travels at constant speed of 100 miles/hour, after two hours the car has traveled 200 miles. I paraphrase ... "it turns out that the area under the curve (where curve is drawn on time / speed axis) is equal to the length of the curve."

I guess the above claim only holds for a constant curve.

Hey sithe and welcome to the forums.

With calculus, the thing that will help you really understand it is when you understand what is changing (the differentials and the derivatives) and how the integral relates to summing those changes.

With arc-length in normal 2D space (the x-y plane you are used to), the way you think about it is that you have ds^2 = dx^2 + dy^2 and then you want how things change so that you can basically 'add up all the changes' which ends up giving you an integral.

Intuitively you can think of integrals as 'adding up changes' and derivatives as 'finding the changes'. Using this result, you can see at least from this standpoint that both are inverses of each (summing changes and finding changes) and this is the fundamental theorem of calculus.

You might ask "what kind of changes are there?", well apart from the normal changes like the arc-length, or standard changes in functions like dy/dx, we can also have geometric changes and these are covered in vector calculus. Once you learn the basics of geometry which include dot and cross products and understand what they really mean, if you understand calculus in terms of what is changing, it will make vector calculus a lot easier if you can see what is changing and how it's changing.

You are also right about the constant example, but remember to think about things in terms of what is changing when you look at the equations because that is how you will really understand calculus and eventually if you choose to, end up deriving your own results on your own.

Good luck.

 

[sithe]

Thank you Chiro ... that's is what I hope to achieve ... I want to understand the concepts at the fundamental level and not just the equations