If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:9:18

Worked example: arc length (advanced)

Video transcript

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 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 purplish 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 little 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 1 plus f prime of x squared DX 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 1 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 I'm going to be let's see X to the third derivative of that is 3x squared so 3x squared 3 over 6 so it's going to be x squared over 2 and then this is 1/2 X to the negative 1 is one way to think about it and so that's going to be negative 1/2 X to the negative 2 negative 1/2 X to the negative 2 power now what is f prime of x squared so it's going to be actually let's just write out what is 1 plus F prime of x squared so it's going to be 1 1 plus F prime of X F prime of x squared is going to be equal to it's going to be 1 plus this squared and so let's see this term right here squared is going to be X to the fourth over four so it's going to be X to the fourth over four now the product of these times two is going to get us negative CX squared times X to the negative two is just going to be one and so it's just going to be negative one half right I just took the product of these two and then multiply that by two yep it's going to just 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 and let's see we have a 1 and a minus 1/2 so we can simplify that a little bit this is going to be equal to X to the fourth over 4 1 minus 1/2 is plus 1/2 plus 1/4 plus 1/4 X to the negative 4 power let me make that a little bit clearer X to the negative 4 power so this just seems a little bit strange and we're going to take a square root of it but maybe we can set this up so that it's a a product of perfect squares because it does look you know X to the fourth and let's see if we can write this in a way that we would recognize how to factor it a little bit better so let's factor out a 1/4 X to the negative fourth power so this is going to be equal to 1/4 X to the negative 4 times so see I factored that out so if you factor 1/4 X to the negative fourth out of this first term and I could color-code them a little bit this first term when you factor this out is going to be X to the 8th power now this term right over here if I factored out a 1/4 it's going to be equal to 2 X to the fourth so this term right over here once again this is the powering through the algebra part so plus 2x to the fourth and then this one is pretty straightforward this is just going to be equal to 1 so plus one plus one and now this is looking interesting because this one well this right over here we could rewrite as 1/2 X to the negative two squared this is equal to that same color 1/2 X to the negative 2 squared and then this over here we could rewrite this as X to the fourth plus let me write it this way this is going to be X to the fourth plus 1 squared right X to the fourth times X to the fourth is X to the 8th 1 squared is 1 the product of them times 2 is 2 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 1 plus f prime of x squared notice I'm just focusing on the algebra so what we've done so far this is 1 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 1/2 X to the negative 2 power x times X to the fourth plus 1 times X to the fourth plus 1 X to the fourth plus 1 I do that right yep or I could actually now that I did all of this to put this in a form that I could actually recognize now I can distribute things back this is going to be equal to this is going to be equal to 1/2 X to the negative 2 times X to the fourth is x squared 1/2 x squared plus 1/2 X to the negative 2 and if we're going to take if we're going to take the ax let me rewrite this so that's 1 plus F prime of x squared but now let's take the definite integral so I'm going to give myself some space to write my DX so we're going to take the definite integral in this case from x equals 1 x equals 1 to x equals 2 so it's the definite integral from x equals 1 to x equals 2 of this DX so it's going to be the definite integral of this from x equals 1 to x equals 2 DX and so this is fairly straightforward the antiderivative of 1 half x squared that's going to be what that's going to be 1 C X to the third and we divide by 3 so 1/2 divided by 3 is 1/6 1/6 X to the third and then this is going to be we're going to increment this is going to be X to the negative 1 we're going to divide by that so minus 1/2 X to the negative 1 power is that right did I do that you have negative 1 when you take it yep negative 2 and then this one yeah that looks good and we're in the homestretch we're going to evaluate it at 2 and at 1 and so we get when you evaluate it at 2 you get 2 to the third which is 8 over 6 minus 1/2 times 1/2 so this is minus 1/4 and when you evaluate it at 1 you're going to have minus so we're going to subtract evaluating it at 1 it's going to be 1/6 minus 1/2 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 4/3 4/3 minus 1/4 minus 1/6 and then we have plus 1/2 now see the common denominator here would be 12 so this is be 4/3 over 12 is 16 over 12 all right and multiply the numerator denominator by 4 minus 1/4 the same thing as 3 over 12 this is minus 2 over 12 and then we have plus 6 over 12 and so this is going to be equal to I think we deserve a little at least little mini drum roll right over here so 16 minus three is going to be 13 minus 2 is 11 plus 6 is 17 so there we have it the length of that arc along this curve between x equals 1 and x equals 2 that length right over there is 17 17 12 and we're done