00:01
We are going to compute the surface area of this function from 1 to e when it's revolved about the x -axis.
00:09
So recall that surface area is the integral, in this case from 1 to e, of 2 pi y, which is 1 .4x2.
00:26
2 .5 .xx times the square root of 1 plus y prime squared.
00:37
So let's compute y prime over here.
00:43
Y prime is x over 2 minus 1 over 2x.
00:52
And then we have to square y prime.
00:56
So we have to square this portion.
01:12
So we get the following integral.
01:14
And now let's focus on this piece first.
01:34
This is equal to 1 plus x squared over 4 minus 2 times x over 2.
01:50
Times 1 over 2x plus 1 over 4x squared.
02:02
And then these x's will cancel, these 2s will cancel, and this gives us 1 half plus x squared over 4, plus 1 over 4x squared, which can be written as x over 2 plus 1 over 2x squared.
02:37
But this square root and that square, they will cancel out.
02:42
So we get that this is the integral, again, from 1 to e of 2 pi, y, which is x squared over 4 minus l and x over 2 times x over 2 plus 1 over 2x dx.
03:18
And so now let's multiply through.
03:20
We get and factor out this 2 pi, or pull this 2 pi out in front of the integral.
03:27
And we get that this is 2 pi times the integral of x cubed over 8 plus x squared over 8x minus x times l in x over 4x.
03:56
Minus l in x over 4x.
04:06
That's dx.
04:10
And then right away we can see that we can simplify this.
04:15
And we get that this is 2 pi.
04:19
Let's also break this integral up into pieces to handle each one individually.
04:25
So that's 2 pi times the integral from 1 to e of x cubed over 8 dx plus 2 pi times the integral from 1 over e of x over 8 dx minus 2 pi times integral from 1 to e of x l and x.
05:01
Dx minus 2 pi over 4 times the integral from 1 to e of l and x over x d x and then let's give a name to each one so we can let this be our first integral we can let this one be our second one and then so on and so forth and then what we can do is just solve each one separately so the first the first one is pretty simple.
05:53
So this is 2 pi times that's x to the fourth over 32.
06:03
So this is e to the fourth over 32 minus 1 over 32, which we'll leave like that for now.
06:23
In the end we'll add all of these together.
06:28
For the second integral, this is 2 pi times x squared over 16.
06:39
But then we have to evaluate that at the limits of integration so that's e squared over 16 minus 1 over 16 and then for the third integral we have minus pi on 2 times and we have to solve this integral which is x times l and x so let's do that over here well in this case we can let you be l and x which i means that e to the u is x, so e to the u, du is dx.
07:49
And so now we get that this is the integral of e to the u times u times e to the u, and then from here we can use the tabular method.
08:11
We want to differentiate u, which is 1, and then 0, and then 0, and then and then integrate e to the 2u, which is one half, e to the two u, and then that's one -fourth, e to the two u.
08:35
And then let's not forget the signs on the side...