![]() And so this is going to be equal to, I think we deserve at leastĪ little mini-drum roll right over here. Alright multiply the numeratorĪnd denominator by four. So this is be four thirds over 12 is 16 over 12. Now let's see the commonĭenominator here would be 12. This is four thirds, four thirds minus oneįourth minus one sixth and then we have plus plus one half. So this is going to be, let's see This is, if we divide, So we're gonna subtract evaluating it at one, it's gonna be one sixth minus one half) And now we just have toĮvaluate these fractions. And when you evaluate it at one, you're gonna have minus 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. Take it, yep negative two and then this one, yep, that looks good. Is that right? Did I do that? Yep, negative one, when you That so minus one half X to the negative one power. And then this is going to be, we're going to increment, this is gonna be X to the negative one. So one half divided by three is one sixth. Let's see it's X to the third and then we divide by three. The anti-derivative of one half X squared that's going to be, what? That's going to be one, So it's going to be theĭefinite integral of this from X equals one to X equals two D X. Integral from X equals one to X equals two of this D X. So we're gonna take the definite integral, in this case, from X equals one X equals one to X equals two. If we're gonna take, Let me rewrite this. This is going to be equal to one half, X to the negative two That I could actually recognize, now I couldĭistribute things back. Or I could actually, now that I did all this to put this in a form X to the negative two power times, times X to the fourth plus one. ![]() Let me just color code it, this is going to be one half And so that's going to be equal to square root of this is, Times X to the fourth is X to the eighth. And this over here weĬould 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. This is equal to, that same color, one half X to the negative two squared. Because this one, this right over here we could rewrite as one half So plus two X to the fourth, and then this one is So this term right over here, once again, this powering Here, if I factored out a one fourth, it's going to be equal to two X to the fourth. This first term, when you factor this out is going to be X to the eighth 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 oneįourth X to the negative fourth out of this first term, and I could color-code 'em a little bit. So let's factor out a one fourth X to the negative fourth power. That we could recognize how to factor it a little bit better. Up so that it's a product of perfect squares, because it does look, you know X to fourthĪnd let's see if we can write this in a way One half plus one fourth, plus one fourth X to This is going to beĮqual to X to the fourth over four. One and a minus one half so we can simplify them a little bit. And then this term right over here squared is going to be positive one fourth X to the negative four. Product of these two and then multiplied that by two, yep, it's just gonna be negative one half. And so it's going to be negative one half. X to the negative two which is going to be one. These times two is going to get us negative, Let's see X squared time Term right here squared is going to be X to the fourth over four. 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. Now, what is F prime of X squared? So it's going to be, Actually let's just write And so that's going toīe negative one half X to the negative two. This is one half X to the negative one is one way to think about it. Over six is going to be X squared over two, and The derivative of that is three X squared. So what is F prime of X going to be? Let's see, X to the third, Need to square it, we need to add one to it, and then we X we just need to figure out what F prime of X is, we Of X we could actually deal in terms of other variables. Lower boundary in X to the upper boundary in X,Īnd this is the arc length, if we're dealing in terms We got arc length, arc length is equal to the integral from the We got a justification for the formula of arc length. The arc length formula correctly, it'll just beĪ bit of power algebra that you'll have to do toĪctually find the arc length. So I encourage you to pause this video and try it out on your own. And so we've already highlighted that in this purple-ish color. ![]() Length along this curve between X equals one and X equals two. And what I want to do in this video is to figure out the arc This right here is the graph of Y is equal to X to the third over six plus one over two X.
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