noc21-ma20-lec27

[Music] [Music] so yes in the last lecture we are discussing about ah how to find the the arc length of a curve we will see that ah not in all the cases we can compute the arc length of the curve but for certain a large section of the curve class of the curve we can find the arc length so lets ah take an intuitive idea about how to proceed as we have seen we had given a curve which is from a to b and then we take the partition of equal length interval and then what we do is that we join the two end points this is x naught f of x naught and x one f of x one so we join the straight line and then we go on like this so basically now what we had observed if we are making our partition smaller and smaller that is a natural definition for the arc length of the curve so now let us see what is the distance between we want to the length of the straight line which joins length of the line joining x naught f of x naught rather i can we can say that f x i minus of one f of x i minus of one and x i f of x i so we know precisely what is the distance what is the length of the straight line so the length of the straight line is equal to square root of x i minus of x i minus of 1 square plus f of x i minus of f of x i minus of one square that is what is length of the straight line ok so now let us assume that our function f is a nice function in the sense that it is differentiable and it satisfies the mean value property so then observe f of x i minus of f of x i minus of 1 divided by x i minus of x i minus of 1 this is equal to some f prime of x i star where x i star belongs to x i minus of one and x i by mean value theorem for nice f right we that means the curve what we are looking at is a differentiable ah function to start with so then we can get this so this we can write this this is the square root of ah let's denote delta of x is equal to x i minus of x i minus of one then this is delta of x square plus f of x i star square and then this is delta of x square now if we take this out then this is going to be square root of 1 plus f prime x i star square times delta x now if we take our partition smaller and smaller then and if we are given that this is f is nice such that f prime is continuous so on and so forth so this is an integrable function so taking the partition small what we are going to get that this converges to a to b square root of 1 plus f prime of x square dx so that is what is going to be the arc length of the curve so let us so this l what we are writing is a to b square root of 1 plus f prime of x square dx this is the arc length of the curve so which means you are looking for the curve which has nice property such that curve y equal to f x which has nice property such that f prime is also continuous then this has no issue in finding the arc length so let us do an easy example find the length of the curve y is equal to 1 plus 6 x to the power 3 upon 2 and 0 is less or equal to x is less or equal to 1. now what is our f of x so f of x is equal to one plus six x to the power three by two therefore f prime or d y by d x so if i calculate d y by d x then this is going to be six into three by two into square root of x so now so arc length is equal to now the interval is from 0 to 1 and then this is square root of 1 plus this is 2 to 9 is 81 x d x now of course by using substitution we know how to compute this so this is ah probably going to give me 1 by 81 then this is 1 2 ah square root of 82 ah now t is ok so i will put it as 82 square root of t dt so this we can easily compute what it is going to be so similarly ah i mean you can you can find the curve ah whenever the curve is given in the equation y equal to f x you calculate f prime and look at the interval in which you are trying to measure the length of the curve and that is going to be essentially through the integration this is a very nice application of the integral so here what we are saying that ah our curve is given by y equal to f x which means it is a graph of a function and this function is nice sometimes it happens that suppose suppose you consider a particle is moving like this in in this fashion so obviously this we are we will not be able to represent it as y equal to f x because this is going to break the rule of straight line so vertical line test this is going to fail so this may not be cannot rather in this example cannot be written as y equal to f x then how how we are going to and estimate or calculate the length of this curve now this for example this what we are giving here we are saying that a particle is moving like this fashion now this particle is moving in this fashion means this what we are observing that it is ah dependent on time given a particular time where the particle is that is how we are going to ah estimate just imagine that if you are traveling on ah on hills so you are going rotating and getting some height with the curve and all but with the time you are making moving the distance and traveling more and more distance given time so now how we are going to compute ah ah that so now it is we are going to represent this suppose this point at a particular time t now this is go in the coordinate plane this is x comma y now this position x comma y they are also function of time so so x and y coordinates of the particle is a function r function of time that is x is equal to f of t and y is equal to g of t and in the time interval you are looking at from a to b let us say from 1 hour to 2 hour or something like that whatever but this is on the interval a to b so this is given by parametric equation that means there is a function f is a function from a b to r and g is also a function from a b to r so all these x and y they are been given in terms of a parameter in this particular example that is the parameter is the time and which is on a fixed interval a b so such curves in general are called parametric curves so any curve given any curve c given by is called a parametric curve ok so for example the simplest example what we can take let us take x equal to cos t and y is equal to sin t then what is going to be the shape of this curve now the shape of this curve as we know if i look at the interval between 0 to t is less than or equal to pi if i want to look at then this is going to be a semicircle or if i make it 2 pi then this is going to be the full circle so now we would like to get the arc length of this parametric curves now for the time being let us assume that let us see some of the curve parametric which is given by the parametric equation can also be by eliminating the parameter one can get that y is some function of f in some cases by eliminating parameter t we get y is equal to some f of x so as you can see that for simplistic because this is what x is equal to f t is given y is equal to g of t is given now suppose our f is so nice let us say the inverse exist ok then in this case if inverse exist then t can we can write it as f inverse of x now this y i can write g of t t is f inverse so here in this case we are getting that g is equal to capital some other it is a function of f so in particular you you know that if f prime is not equal to 0 then you can you can so that this is 1 1 and if f prime is also a continuous function you can show that that is going to be ah if inverse is going to exist and so you can do so some condition on this f can always give you lan us into this situation where we can write y equal to f of x so let us try to see that if we have in this situation what we are going to get as the arc length of this curve ok so now this what we are writing so this is ah y is g of t this is equal to f and x is f of t so this is f composed of t so if i take the derivative with respect to t then this is d y by d t is g prime of t this is f prime of f of t into small f prime of t ok so this implies capital f prime of x this is equal to g prime of t by f prime of t if f prime of t is not equal to zero so in in our notation what is this capital f this is d y by d x this is equal to g prime of t y equal to g t so this is d y by d t divided by d x by d t provided if d x by d t is not equal to zero this is what we are getting that this is the if we can write y in the form of ah x then d y by d x is equal to f d y by d t divided by d x by d t now the slope of the tangent of the curve y equal to f of x then slope of c in this particular where y is written as y equal to f of x at x f of x is f prime of x right so now we can find the slope of the tangent at the curve now for example a curve c is defined by the parameter parametric equation rather y is equal to t q minus of three t so now this is fine the tangent at three zero now three 0 means i need to find what are the thing this is the point 3 0 so therefore t [Music] into t square minus of 3 this is equal to 0 this implies that t equal to 0 or t equal to plus or minus of root 3 right so now if t is equal to 0 x is equal to zero that but we are interested in three zero so t is equal to plus or minus of root three now we want to find the slope that is d y by d x which is nothing but d y by d t divided by d x by d t so this we can just compute it in this case ah this is at at t equal to square root of 3 then what we are going to get is that and d y by dx if we look at at 3 equal to minus of square root of 3 this is going to be if we calculate this is what it is going to ok so now let us answer in this situation let us try to get what is what is going to be the arc length so suppose c is a parametric curve x is equal to f of t y is equal to g of t and alpha is less or equal to t is less or equal to beta t that means t increases alpha to beta and denote f of alpha this is equal to a and f of beta this is equal to b therefore in this case what is your l what we want to write this is we are trying to write a to b square root of 1 plus d y by d x whole square right dx now this is equal to a to b now if i change it change the parameter to t making the substitution then what do i what do we get we get d y by d x equal to d y by d t square divided by d x by d t square now d x is what d x by d t x prime by substitution rule and then this is d t now this is a because we are putting a to b is going to be t will vary from alpha to beta so this if we compute then this is going to be alpha to beta square root of d x by d t square plus d y by d t square then this is d d so this is what if c is expressed in terms of the capital f even even in the case if c is not expressed by the form of y equal to capital f of x ah by polynomial approximation we can get this formula so let us let me ah write it down the statement so this is the theorem if a curve curve c is described by parametric equation x equal to f t y equal to g of t alpha less or equal to t less or equal to beta where f prime and g prime are continuous and c travels exactly once at t increases from alpha to beta then the length of is l is equal to integral a to b ok so this is irrespective of whether y equal to capital f of x we are able to eliminate ah t ah or not ah this is what is the statement so let us do ah some concrete example ah let us say simple what we have started with cos t y equal to sin t and then t belongs to zero two pi then l is equal to dx by dt dx by dt square is equal to 0 to 2 pi square root of sine cos t is minus sin t so this is square is sin square t plus d y by d t is cos t square is cos square t d t which is equal to two pi that is what the length of the curve what we know now another quick example so let x is equal to r t minus of sin t and y is equal to r 1 minus of cos t and again the interval is from zero to two pi so you can compute the d x by d t square plus d y by d t square if you compute that by taking the derivative what you are going to get easily you can see that 4 square sine square t by 2 so therefore the l what you are going to get is nothing but 0 to 2 pi this is equal to square root is 2 r sine t by 2 dt and which if now sin t by 2 we know the anti-derivative we can compute so you are going to get h r so this is even if we do not have kind of y equal to f x as a graph of a function but any curve which is given by the parametric representation that means x equal to f t y equal to g t then basically the length is going to be ah arc length is a to b square root of f prime t square plus g prime t square d t that is what is going to be the arc length of the parametric curve ok so then in the next lecture what we are going to see is that sometimes you know the curve can be given by something called a polar coordinate so how you are going to do analysis compute arc length the area if our curve is given by polar coordinate thank you [Music] do [Music] you