So we have to square root of two X to the one half and we want to square it. The second half second step is that we need to square this. So the three canceled in this four to just becomes to squarer too x to the one half. We're going to bring down the three halves X and we're going to subtract one from the three halves, leaving us with one half and the exponents and then the minus one is gonna integrate to zero. X is equal to So we're going to keep four square were to to over three. The first thing we're going to do is find that derivative. So again we're going to follow the steps. So the equation for the length of a curve thanks is equal to the integral from a to B of the square root of one plus b Y d x squared dia. This is the equation for the curve y equals four haps, four square to over three extra, three house minus one, and X is from zero toe one. Okay, for this example, were asked to find the arc length of this curve were asked from the length of the curve. And now on the next video, we'll do the introduction to surface area. So this is an introduction toe are cling. So the length of a to be square root one plus dx dy y So the derivative of the function squared and then d x. So for the final answer, the arc length which we call l is equal to the integral from A to B, where a is where your line your curve starts. You are approximating the length of this curve using line segments, and then you use the mean value fear, um, and taking the limit. We can rewrite as DX brief proof of how we end up with the arc Link formula. We'll just call it X Square in Delta X K. Plus, we have this derivative which is f prime. This changes to an integral square root one. So now we're gonna take the limit of the summation, and this is going to be equal to the integral now from A to B because we're taking the limit. So we have this and this is equal to when we rewrite it and we take the limit. And what we know about that Delta X k well, this it's squared, but the squaring square root are getting canceled. He's okay, squared times, and we still have that Delta X K. And with some algebra you get, This is equal to the summation of the square root of one plus f prime. And now we can distribute the square and do some factoring. But then we have plus, if crime C sub k So a derivative times that change in X okay, squared. Okay, clicking this in, we get the summation Square root, Delta X squared. Why? Which is to that F prime cease of K. We can use the mean value serum, and we let that Delta X be equal to ex prime. So this is just walking you guys through a brief proof of how we come up with the equation. So to find length, it's the change in the X variable squared, plus the change in the Y variable squared. So from N which is really six in our case of and now we're going to actually write out the equation for length. So if you add up the length of each of these line segments, you get the approximate length of the curve. So the length of this curve is approximately equal to so the length of this curve is approximately equal to the summation of the different L case, where K goes from one in our case, Thio end. We have the line segment L two to l three 03 to 4 of 14 055 l six. So let's say we have the line segment l want l two. If we draw these lines straight lines connecting the curve, various points, so we'll label each of these. So what arc length is doing is saying if we have this curve, we can approximate the length of the curve. So starting with our plank, let's say we have this curve.
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