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How to calculate length?

In this post I want to find the length of this curve from here to here how can we do it of course it's a curve so I can I just use it over hmm we can use the daughter along with calculus then attack you okay let me show you because imagine this is a daughter you have to just place the loot right here and what you get a tangent line excellent so here's a teal maybe I will just pick a point for demonstration right here and I'm just going to join it attention step in like this and this right here is supposed to be really small but if I make you so small nobody can see it.

I will just have to make it big and the idea is that this right here this length we can calculate it and this length it's going to be really really close to this portion of the mark in blue if you make this really really small and the idea is that we are using the red part to approximate the blue part and then we just add up all the stock red part then we will get the art legs that's pretty much the idea all right for this right here I will call it to be the L some books copy the little s doesn't really matter well here's the deal I will actually look at this and I'll go horizontally like that because this right here is the small change amount in the x-values that's DX and then I will go here and up and that will be dy a small change in the Y values.

why do we need VX and V Y well because as you can see this is where we can do that the profit here and also by doing that we come out with a little triangle little red triangle so we get to use the Pythagorean theorem so with that being said we know that by Pythagorean theorem PL squared is equal to this square plus that squared so let me just write it down as DX square plus the Y's we're here and of course I want to get deal I can just take the positive square root on both sides.

Now here is the deal for the our thanks some books they put d2s for the notations I like to use capital L but since this is my first video especially we're talking about proof I was to spell everything else anyway here's the deal once again this is the DL and this is supposed to be really close to the length of this arc in blue and just need to add up such detail deal so for the art length from here to here it's pretty much that I will just have to do DL and then add them up and ideas of course just integration unfortunately though I cannot put down any number here.

I don't know too much about the air world so we have trouble and how can we fix that of course we have to refer by DL to be this and you see that we have the DX here and also the dy here and then of course they are being squared instead of the square root in fact there are some algebra that we all have to do and it depends on the situation yeah two situations the first situation is that maybe this curve right here is defined as Y to be a function of X so I will just write down when we have y to be a function of X so radius y equals f of X then in that case we do the following I'm going to look at this expression and I will factor out the DX squared all right let me show you so this is going to be I will just put them all now yes the integral of course but for the DL once again I will factor out a DX squared instead of the square root so let me open a square root first and I will just put down the DX square first like that and we will have well originally was DX square by franticly already so that would be 1 and we added with this didn't have the X there's a DX squared so I will have to define.

I will have to put down dy Square and then divided by the DX squared like that now you see we have the DX square times this quantity instead of the square root so it's pretty much a square root of this times the school of that from here you see the square and a suru cancel and we just get a DX and we can put that at the end of the integral and I will read it down for you guys this is the integral and this is going to give me just a regular DX I will put that at the end right here and for the integrand it's just the square root of 1 plus and notice that they are both to the second power so of course we can read yes dy DX and then you Square and here's the deal for the dy DX you are going to look at this equation differentiate Y with respect to X namely you to the derivative right here with respect to X get a derivative put here square that add a 1 open as putting the square root DX and because when the ex-world go back to the picture look for the x value and i'll just call the x value to be a to be like that.

You know you will have to go for a to be and that's the X values so this is the arc length boom law the first version and you use this when you have Y as a function of X on the other hand if you were given X as a function of Y really similar to the previous video we may have this situation as well and in this case for the DL part I'll be looking at this and then factoring out that the Y square instead of the DX square so it's pretty much check the APIs of that and now let's get to work we write down L is equal to integral and for the DL you open the square root first and then we will factor out that the Y square put it here first and then from here we get this term it's going to be DX square over the Y square and then we add in words this is going to be just a 1 and do not put down tyo right here because we don't have any like this we didn't have any look at this is square root of dy square squares where we can so so this ey at the end .

we have the square of this as the integrand so in the end you see that L is equal to the integral and the integrand is square root and let me put down the 1 first and we will be adding with this DX square over the us square we can write es DX dy and then you square that and this right here is the D Y at the end and you see we are in the wide world so look back to the picture we will have to figure out the Y values here up to here let's just cut out to be C and D so we will have to know Y is going from C to D and DX dy is that you just look at this equation here differentiating X with respect why do the derivative here with respect to Y and put it here and that's pretty much it so what's up being said these are the two formulas for the are lengths depending on the situation and as always you know let me know if you guys have any questions and that's it

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