0:00
Hi there.
00:01
Today we're going to look at the surface area you get when you take the function y equals a half times the natural logarithm of 2x plus the square root of 4x squared minus 1 and rotate it about the y -axis from a half 0 to 17 n of 2.
00:20
Now because we are rotating this about the y -axis we need to find the inverse of this function.
00:27
That is we need to find x as a function of y.
00:30
So the first step in this is to get rid of the half and get rid of the natural logarithm.
00:36
So we'll multiply by two and exponentiate.
00:39
So we have e to the 2y is 2x plus the square root of 4x squared minus 1.
00:48
Now consider e to the minus 2y, which is 1 over 2x plus 4x squared minus 1 and consider taking their sum.
01:01
To the 2y plus e to the minus 2y.
01:05
So this will be 2x plus 4x squared minus 1.
01:12
And if we square this and divide by itself, we have in essence multiplied by 1.
01:21
So this is fine.
01:22
However, by doing this, we're able to find a common denominator.
01:28
So we can treat these together.
01:31
Now if we develop this square, we'll have 4x squared plus 4x 4x squared minus 1 square root.
01:42
And then we'll have the square of our square root.
01:45
So plus 4x squared minus 1 and then we'll have plus 1 from divide by 2x plus 4x squared minus 1.
02:00
Now if we notice the 1s at the end cancel and then here we have an 8x squared.
02:07
So we can factor out 4x from the two remaining terms and we'll see we get 2x plus 4x squared minus 1 square root over 2x plus 4x squared minus 1 square root and that's just 4x.
02:27
Therefore our inverse is g of y which is x which is a quarter e to the 2y plus 2y plus to the minus 2y.
02:42
And now for those of you who have experience with hyperbolic functions, you can recognise this as a half kosh of 2y.
02:58
I'm not going to use that method here.
03:01
However, those of you who have covered hyperbolic functions can do that within this framework.
03:10
Now, we need to use g prime of y.
03:15
This is because the surface area formula is 2 pi times the integral between our limits, g of y times a square root of 1 plus g prime of y squared, whole square root d, y.
03:31
So if we look at g prime of y, this is a half times e to 2y minus e to the minus 2 y.
03:44
If we square this, g prime of y squared, we'll have a quarter, e to the 4y, plus minus 2 plus e to the minus 4 y.
04:07
And then if we take g prime of y squared plus 1, which is this whole term, we can see that this will, if we add 1, this is the same as adding 4 over 4.
04:28
So if we bring that in here, this 4, it will change this to a plus 2...