00:01
In this problem, we are provided with the curve y equals to x raised to the power 6 plus 8, the whole divided by 16 x squared.
00:14
And we are asked to find out the arc length when x varies from x equals to 1 up to x equals to 2.
00:27
The formula for arc length is given by l equals to integral a to b square root of 1 plus dy over dx the whole squared dx where a and b stand for the limits which is clearly 1 and 2 respectively in this case.
00:55
So now let us find out the value of the integrant.
00:58
We consider the given curve and we simplify it so we have x raised to the power 6 over 16 times x squared plus 8 over 16 times x squared here 8 and 16 get cancelled twice and in the first term x squared and x power 6 get cancelled so we get x power 4.
01:22
So this implies that y equals to x raised to the power 4 over 16 plus 1 over 2 times x raised to the par negative 2.
01:35
Now let us differentiate this with respect to x so we have dy over d x equals to the derivative of x raised to the 4 is 4 times x cubed the whole divided by 16 plus 1 over 2 times the derivative of x raised to the power negative 2 is negative 2 times x raised to the power negative 3.
01:58
Simplifying this, 4 and 16 get cancelled 4 times, 2 and 2 get cancelled.
02:04
So we obtain x raised to the power 3 over 4 minus 1 over x cube.
02:14
Next we square on both sides.
02:19
So we have dy over d x the whole squared equals to squaring on the right hand side by making use of the identity, a minus b the whole squared equals to a squared minus 2 times a times b plus b squared.
02:38
We have x raised to the past 6 over 16 minus 1 over 2 plus 1 over x raised to the past 6.
02:55
Next, we add 1 on both the sides.
03:00
So we have 1 plus d .y over dx the whole squared equals to x raised to the past 6 over 16.
03:11
Negative 1 by 2 added with 1 gives us positive 1 over 2 plus 1 over x raised to the pass 6.
03:19
Making use of the identity a plus b the whole squared which equals to a squared plus 2 ab plus 2 ab plus b squared we can rewrite the right hand side as x cubed over 4 plus 1 over x cubed the whole squared...