00:01
So here given the formula of y, we want to know the arc length of y on the interval from 2 to 8.
00:09
So recall the formula for arc length is s equals to the integral here is from 2 to 8, and square root of 1 plus the derivative of y squared x.
00:23
So first we will calculate the derivative of y, which is 3 times 3 over 2 times x2.
00:34
Of 1 over 2 minus 1 2 x3 power of negative 1 over 2.
00:40
This is just square root of x over 2 minus 1 over 2 times square root of x.
00:47
Then let's look at the formula of 1 plus derivative y square.
00:55
This is equals to 1 plus square root of x minus 1 over 2 square root of x square.
01:05
But then notice this one here, we can write this as 4 times square root of x over 2 times 1 over 2 times square root of x.
01:20
And then we have the same square here.
01:27
Therefore, we can plug in this term and then obtain this expression just being square root of x over 2 plus 1 over 2 times square root of x of 2.
01:42
Square.
01:43
So this expression is really a perfect square, which can make the integral much more easier...