00:01
So here we're going to find the area of the surface generated by rotating this curve about the x -axis between the bounds of x -equals 1 and x -quels 3.
00:10
And to do that, we're going to be using our area of a, the surface area of a revolution solid or a revolved curve, however you want to say it.
00:23
And so first thing we're going to need is to get our derivative in terms of x.
00:28
So we're going to use some quotient rule here.
00:33
This is going to be derivative of the top times the bottom minus derivative of the bottom minus the derivative of the bottom minus the top all over the bottom squared.
01:06
So let's do some simplifying.
01:08
We got 48 x to the 7 minus 16 x to the 7 minus 32x all over 6.
01:25
For x to the 4 let me fix this 4 it's bugging me so let's go ahead and take out a 32 from the top and the bottom after we subtract these two we get x to the 7 minus x over 2x of the 4 so here's what we're going to use for our area integral and so don't quite have enough space i'm just go ahead and start down here so we're going to take the integral from 1 to 3 times our y or our f of x x to the 6 plus 2 over 8x squared.
02:11
And now we're going to plug in our square root of 1 plus the derivative squared.
02:20
X to the 7 minus x over 2x of the 4.
02:27
And this is going to be dx.
02:30
Alright, so first thing i'm going to do is expand this inside.
02:39
Let's do the whole thing.
02:40
One second.
02:41
I'm going to expand this entire inside the radical and see what i can get.
02:48
So let's put a star here and a star here.
02:52
So if we expand the x7 minus x over 2x to the 4 squared, we'll get x to the 14 minus 2 x to the 8 plus x squared all over 4 x of the 8.
03:18
So this is going to be, and then we have a plus one...