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Worked example: arc length (advanced)
- [Voiceover] This right here is the graph of Y is equal to X to the third over six plus one over two X. And what I want to do in this video is to figure out the arc length along this curve between X equals one and X equals two. And so we've already highlighted that in this purple-ish color. So I encourage you to pause this video and try it out on your own. And I'll give you one hint. Assuming that you apply the arc length formula correctly, it'll just be a bit of power algebra that you'll have to do to actually find the arc length. So I'm assuming you've had a go at it. Let's work through it together. So a few videos ago, we got a justification for the formula of arc length. We got arc length, arc length is equal to the integral from the lower boundary in X to the upper boundary in X, and this is the arc length, if we're dealing in terms of X we could actually deal in terms of other variables. And so it's going to be one plus F prime of X squared D X. So for this particular F of X we just need to figure out what F prime of X is, we need to square it, we need to add one to it, and then we have to take the square root of that. So let's do all of that step-by-step. So what is F prime of X going to be? Let's see, X to the third, the derivative of that is three X squared. So three X squared, three over six is going to be X squared over two, and this is one half X to the negative one is one way to think about it. And so that's going to be negative one half X to the negative two. Negative one half X to the negative two power. Now, what is F prime of X squared? So it's going to be, Actually let's just write out what is one plus F prime of X squared? So it's going to be, So one, one plus F prime of X, F prime of X squared is going to be equal to it's gonna be one plus this thing squared. And so, let's see, this term right here squared is going to be X to the fourth over four. It's gonna be X to the fourth over four. Now the product of these times two is going to get us negative, Let's see X squared time X to the negative two which is going to be one. And so it's going to be negative one half. Right, I just took the product of these two and then multiplied that by two, yep, it's just gonna be negative one half. And then this term right over here squared is going to be positive one fourth X to the negative four. One fourth X to the negative four. Now let's see we have one and a minus one half so we can simplify them a little bit. This is going to be equal to X to the fourth over four. One minus one half is plus one half plus one fourth, plus one fourth X to the negative four power. Let me make that a little bit clearer. X to the negative four power. So this seems a little bit strange. And we're going to have to take the square root of it. But maybe we can set this up so that it's a product of perfect squares, because it does look, you know X to fourth and let's see if we can write this in a way that we could recognize how to factor it a little bit better. So let's factor out a one fourth X to the negative fourth power. So this is going to be equal to one fourth X to the negative four times, let's see I factored that out, so if you factor one fourth X to the negative fourth out of this first term, and I could color-code 'em a little bit. This first term, when you factor this out is going to be X to the eighth power. Now this term right over here, if I factored out a one fourth, it's going to be equal to two X to the fourth. So this term right over here, once again, this powering through the algebra part. So plus two X to the fourth, and then this one is pretty straight forward. This is just going to be equal to one. So plus one. Plus one. And now this is looking interesting. Because this one, this right over here we could rewrite as one half X to the negative two squared. This is equal to, that same color, one half X to the negative two squared. And this over here we could rewrite this as X to the fourth plus, so we write it this way, this is going to be X to the fourth plus one squared. Alright, X to the fourth times X to the fourth is X to the eighth. One squared is one. The product of them times two is two X to the fourth. So this over here is the same thing as that. And so now if we wanted to take the square root of both sides. So if we wanted to say the square root of one prime plus X squared. Notice I'm just focusing on the algebra. So what we've done so far this is one plus F prime of X squared. Now we want to take the square root of that. So that's going to be the square root of this. And so that's going to be equal to square root of this is, let me just color code it, this is going to be one half X to the negative two power times, times X to the fourth plus one. Times X to the fourth plus one. X to the fourth plus one. I did that right, yep. Or I could actually, now that I did all this to put this in a form that I could actually recognize, now I could distribute things back. This is going to be equal to one half, X to the negative two times X to the fourth is X squared. One half X squared plus one half X to the negative two. And if we're gonna take, if we're gonna take, Let me rewrite this. That's one plus F prime of X squared. Now let's take the definite integral. I'm going to give myself some space to write my D X. So we're gonna take the definite integral, in this case, from X equals one X equals one to X equals two. So this is the definite integral from X equals one to X equals two of this D X. So it's going to be the definite integral of this from X equals one to X equals two D X. And so this is fairly straight forward. The anti-derivative of one half X squared that's going to be, what? That's going to be one, let's see it's X to the third and then we divide by three. So one half divided by three is one sixth. One sixth X to the third. And then this is going to be, we're going to increment, this is gonna be X to the negative one. We're gonna divide by that so minus one half X to the negative one power. Is that right? Did I do that? Yep, negative one, when you take it, yep negative two and then this one, yep, that looks good. And we're in the homestretch. We're gonna evaluate it. At two and at one. And so we get, when you evaluate it at two, you get two to the third, which is eight over six minus one half times one half. So this is minus one fourth. And when you evaluate it at one, you're gonna have minus so we're gonna subtract evaluating it at one, it's gonna be one sixth minus one half) And now we just have to evaluate these fractions. So this is going to be, let's see This is, if we divide, this is four thirds, four thirds minus one fourth minus one sixth and then we have plus plus one half. Now let's see the common denominator here would be 12. So this is be four thirds over 12 is 16 over 12. Alright multiply the numerator and denominator by four. Minus one fourth, the same thing as three over 12. Now this is minus two over 12. And then we have plus six over 12. And so this is going to be equal to, I think we deserve at least a little mini-drum roll right over here. So 16 minus three is going to be 13 minus two is 11 plus six is 17. So there we have it. The length of that arc along this curve. Between X equals one and X equals two. That length right over there is 17, 17 twelfths. Were done.