[MUSIC].
We defined arc length using an integral. The integral that we'll be using is this
one. I'm going to be integrating from a to b
the square root of 1 plus the derivative squared dx.
Let's apply this to a specific problem. Let's compute the arc length of a little
piece of a graph. So let's look at the graph of y squared
equals x cubed and figure out the length of this arc from 0,0 to 1,1.
Now I could rewrite this equation. Instead of writing y squared equals x
cubed, I could write y equals x to the 3 halves power.
We'll start by differentiating. So because of this, I can write dy, dx is
3 halves x to the 1 half power. We'll put that derivative into the
formula for arc length so I'm going to be integrating.
X goes from 0 to 1, the square root of 1 plus this derivative squared.
So, I'll write 3 halves, x to the 1 half squared dx.
Now, I just want to compute that integral.
But how? Well, I can rewrite the integrand.
This is the integral from 0 to 1 of the square root of 1 plus 3 halves squared is
9 4ths. And x to the one half squared, I'll write
x, dx. Just have no fear, keep going.
Well, I could use substitution. u equals 1 plus 9 4ths x so that du is 9
4ths x or alternatively, dx is 4 9ths du. And that means this integral, it's the
same as the integral of square root of u times 4 9ths du and remind you bounce of
integration. Well an x is 0, u is 1 and when x is 1 u
is 14 4ths. Alright, now I just have to write down an
anti derivative here, well I could write this as 4 9th, u to the 3 halves over 3
halves that's an anti-derivative for square root 2, evaluated at 1 and 13 4th.
So I'm just going to plug in 13 4th plug in 1, take the difference.
Okay, so this is 4 9th, I write, times 2 3rd, instead of dividing by 3 halves.
Times 13 4th. To the 3 halves power minus 4 9th times 2
3rd times 1 to the 3 halves power and I got to simplify this little bit more even
lets see here. 4 times 2 is 8, okay that's also 4 to the
3 halves power. So I can cancel all of these things.
And then I can simplify gotta do some arithmetic here.
So I've got what, 13 to the 3 halves power divided by 9 times 3 is 27, minus,
this is 8 27ths just times 1. And, if you don't like the 3 halves here,
I could write this as 13 times the square root of 13 over 27 minus 8 over 27.
This, is the length of that arc. We did it, but it worked out almost too
well. Alright that's the sad thing about arc
length calculations. Because the arc length formula involves
that square root of 1 plus a derivative squared, it's extraordinarily unlikely.
And if you just give me some random function, I'm going to succeed in being
able to evaluate that integral. The sorts of things we can calculate the
arc length of are pretty special sorts of things.