00:01
For this problem we are to find the arc length of y that's equal to the natural log of x minus the square root of x squared minus 1.
00:09
For x values in the closed integral 1 to the square root of 257.
00:14
And we are given a hint that is to integrate with respect to y.
00:20
Now since our function is written as a function of y in terms of x, we have to rewrite this as a function of x in terms of y.
00:30
That using properties of logarithm e raise a power of y is equal to x minus the square root of x squared minus 1 so e raised to y plus e raised to negative y that is the reciprocal of x minus the square of x squared minus 1 this is equal to x minus square root of x squared minus 1 plus 1 over x minus square root of x squared minus 1 this is equal to 2x, which means that x is equal to one half of e raised to y plus e raised to negative y.
01:15
And since our interval is an interval in terms of x, we have to rewrite that as well.
01:22
So if x is in the interval 1 to the square root of 257 and y is equal to the natural log of x minus square root of x squared minus 1, then for x that's equal to 1, we have y equal to the natural log of 1 minus square root of 1 minus 1.
01:47
This is equal to 0.
01:49
And x is equal to the square root of 257.
01:51
It will give us y that's equal to the natural log of square root of 257 minus 16.
01:59
Note that this value is less than 0.
02:02
So when using this as limits for the integral, x equals square to 257 must be a lower limit...