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Current time:0:00Total duration:11:46

Video transcript

what I want to do in this video is come up with the formula for the arc length of a curve that's defined in polar coordinates so if this curve right over here is R is equal to f of theta how do we figure out the length of this curve between two Thetas say between theta is equal to well let's say in this case it looks like between theta zero radians and say PI over two radians but two between any two bounds for our theta so the way we're going to do this and have any point you get excited or inspired you definitely should pause the video and see if you can run with the formula for arc length when you're dealing with something in polar form but the way that we're going to tackle it is the exact same way that we tackled arc length when we were dealing with standard rectangular coordinates so let's take a little small section of the arc length let's take a little small section I'm going to blow it up so let's call this right over here this is our infinitesimally but this is our infinitely small or infinitesimal sized our piece of our arc length I'm going to call it D s and obviously this is a lot bigger than maybe we would imagine when you think of infinitesimal but then if you integrate together all of the DS's if you integrate together all of the D s is then you're going to have you're going to have the length of the actual curve that you care about so we could say that the length is going to be all of the DS's integrated all of the infinitely the infinite sum of all of these infinitely small all of these infinitely small DS's now to actually put this in terms that we can relate to in terms of RS and Thetas I'm first going to relate this to X's and Y's and then relate the RS and Thetas to X's and Y's which we have seen before when we've converted between polar form and rectangular form so we know that this D s is going to be equal to our infinitely small change in X squared so if this is our going from this point to that point that's our change in our arc length but this distance right over here would be our change in our X all right that is DX and I'm writing everything as differentials which is a little bit mathematically we could say hand-wavy er loosey-goosey but it gives you a good conceptual understanding of where this comes from you could if we were a little bit more precise we could take DX that we can do Delta X's and then eventually take limits and all the rest but I'll just go with this because it makes a little bit at least for my brain a little more conceptual sense so that's if that's our change in X when we go from that point to that point then this is our change in Y when we go from this point to this point dy and we've seen this before would we got our justification for the arc length form in rectangular coordinates we could say that D s is going to be equal to the square root of DX squared DX squared plus dy squared and this just comes straight out of the Pythagorean theorem plus dy squared and then if we could integrate these then we're kind of in the same place but how do we how do we get these in terms of RS and Thetas well to do that we just have to remind ourselves what X is in terms of aren't eight and what Y is in terms of our theta so X we know is going to be equal to R cosine of theta and we first saw this when we just first were going back and forth between polar and rectangular coordinates and Y is going to be our sine of theta and now we can use this to say what D X and what D Y are going to be DX is then going to be equal to and we have to remember that R is going to be a function of theta so actually let me write it this way let me just rewrite it so X we could also write it as f of theta times cosine theta and Y is equal to f of theta times sine of theta so now what's DX DX is going to be this is just go we're just going to apply the product rule here it's going to be F prime of theta derivative of the first expression times the second one times cosine of theta plus the derivative of the second one well the derivative of cosine of theta is negative sine of theta so we'll see minus sine theta minus sine theta times first expression so f of theta that was just the product rule that's our DX and then of course D theta D theta another way you could set if you if you treated these differentials like numbers you could divide both sides by D theta you would have the derivative of X with respect to theta is this business right over here so those are equivalent equivalent statements and let's do the same thing for D Y so dy same again by the product rule is going to be F prime F prime of theta times sine of theta plus F of theta times the derivative of sine of theta which is just cosine of theta cosine of theta and now if we want to figure out what D s is we're going to take the sum of DX squared and dy squared so let's do that so D X DX squared is going to be equal to we just need to square all of this business so I'm just going to square this and then multiply that times D theta squared so that's going to be equal to this is going to be F prime of theta squared cosine squared theta minus 2 times the product of these minus 2 times F prime of theta f of theta cosine theta sine theta and then this one squared so negative times negative so plus F of theta squared sine squared sine squared theta so that's DX squared and then of course we have the D theta well I'm not done yet then we have the D theta squared and now let's figure out what dy squared is so dy squared is going to be equal to well this term squared Oh F to forget this dy is going to have a D theta at the end don't want to forget that and so over here this is going to be F prime of theta squared sine squared sine squared theta and then 2 times the products of these so plus 2 times F prime of let me f prime of theta F of theta it's a little bit hairy but we'll see in a few seconds it's going to click it's going to clean up nicely F of theta cosine sine cosine theta sine theta and then we just want to square this Plus F of theta squared cosine cosine squared theta and then D theta squared D theta squared now let's add these two together so let's add them together and what are we going to get so if we add DX squared and dy squared we're going to get so DX squared plus dy squared plus dy squared is equal to so over here we have cosine squared theta times F prime of F prime of theta squared and then sine squared theta times F prime of squared F prime of theta squared so we can factor out an F prime of theta squared so it's going to be equal to so if we factor these characters out it's going to be F prime of theta squared x times cosine squared theta cosine squared theta plus sine squared theta plus sine squared theta and we see that that's going to simplify nicely that this is just going to be equal to 1 that's just one basic trig identity and then these middle two terms actually cancel out this is negative of this so these two cancel out and then over here we can factor out an F of theta squared so we could factor out an F of theta squared so it becomes plus F of theta squared times sine squared theta sine squared theta plus cosine squared theta plus cosine squared theta well that simplifies nicely this is just going to be equal to one and then we have these and then this D D theta squared is multiplied times everything so so everything right over here is going to be so times D theta D theta squared you can almost view these as the coefficients on D theta squared we added those two coefficients so this is going to clean up nicely now so this simplifies as DX squared plus dy squared plus dy squared is equal to is equal to F prime of theta squared plus F of theta plus F of theta squared and then all of that times let me do this in a new color actually all of that times D theta squared actually that I'm going to use that color I'll use the magenta all of that times D theta squared D theta squared now we know that D s is going to be the square root of this so let's write that so D s is equal to the square root of this which is equal to the square root of this which is going to be equal to well we can factor out a the squibs the square root of d theta squared is just going to be d theta so we could just take that out and we are left with we are left with F prime of theta squared F prime of theta squared plus F of theta squared plus F of theta squared and now we took it taking a D theta out of the radical if you put it in as d theta squared you take it out it's going to be d theta D theta so this is interesting so if we wanted to integrate them so if you wanted to integrate this if you want to integrate this if you want to integrate this you just want to D it integrate this right over here and you would integrate it from your starting theta maybe we could call that alpha to your ending theta beta and just like that we have given ourselves a reasonable justification occasion or hopefully a conceptual understanding for the formula for arc length when we're dealing with something in polar form if you have R is equal to f of theta you find what f prime of theta is or you could think of it as the derivative of R with respect to theta square that add that to F of theta squared take the square root and then integrate with respect to theta from out from alpha to beta and so this right over here our arc length is going to be equal to this right over here in the next few videos we will actually apply this