00:01
We're given a curve and we're asked to find the exact length of this curve.
00:05
It has the equation y equals 1 minus e to negative x, where x lies in the interval 02.
00:23
Given this equation, it follows that y prime is equal to the opposite of negative e to negative x, which is simply either than negative x.
00:36
And therefore 1 plus y prime squared is 1 plus e to the negative 2x and therefore the length of this curve this is the integral from 0 to 2 of square root of 1 plus e to the negative 2x d x now here we'll want to make the u substitution u equals e to the negative x then it follows is that du will be negative e to the negative x, dx, which of course is the same as negative u dx.
01:39
And so we have the integral of 0, which is e to the 0, which is 1, to u of 2, which is e to the negative 2, of the square root of 1 plus and e to the negative 2 is the negative 2 is the 2 of 1 plus and e to the negative 2x is the same.
02:01
As e to the negative x squared, or 1 plus u squared.
02:06
This is our first substitution times.
02:11
And we have a dx is the same as negative 1 over u, du...