Expert: Socrates - 1/24/2004

Question
I know this is easy but its stumping me.
My book has the answer as 6+ln(2) but I can't see how its done. Please help.

Find the length of the curve
y=(1/8)(x*x)-ln(x) for 4<x<8

I'm studying for the Math GRE. Do you know any resources that have tons of good questions and answers?  I have the course by The Princeton Review but I want more sample questions.  I've been using calculus textbooks. But if I can't solve the problem the book only gives the answer not the method and that's a problem.

Answer
I really can't give a specific reference to prepare for the GRE. I remember when I took it , I went to a bookstore like Barnes and Noble and got a study guide that had many practice exams . Doing these is the best way to prepare for any standardized test.

For the curve length:

Use the formula S ( 1 + y'^2 )^1/2

In your problem ,

y' = (x/4 - 1/x)

y'^2 = x^2/16 + 1/x^2 - 1/2

1 + y'^2 = x^2/16 + 1/x^2 + 1/2

1 + y'^2 = (x^4 + 8x^2 + 16)/16x^2

1 + y'^2 = (x^2 + 4)^2 / 16x^2

( 1 + y'^2 )^1/2  =  (x^2 + 4)/4x

( 1 + y'^2 )^1/2  =  x/4 + 1/x

S ( 1 + y'^2 )^1/2  = x^2/8 + ln x

The limits are 8 and 4 , so find the difference

64/8 + ln 8 - 16/8 - ln 4 =

8 + 3ln 2 - 2 - 2ln 2 =

6 + ln 2

Which agrees with the answer you have given.