Finding the length of a curve

[rappa]

Question: Find the length of the curve 24xy = x^4 + 48 between (2, 4/3) and (3, 43/24). Answer is supposed to be 9/8.

This is the formula (I think): square root(1 + (dy/dx)^2 dx)

I tried to integrate and I did it like this:

dy/dx = 3/24 x^2 - 2x^-2 or 3/24 x^2 - 2/x^2

then: square root(1 + (3/24 x^2 -2x^-2)^2)

= square root(1 + (1/64 x^4 - 4x^-4))

now integration:

(1 + (1/64 x^4 - 4x^-4))^(1/2)

= (2/3) (1 + (1/64 x^4 - 4x^-4))^(3/2) (x + (1/320 x^5 + 4/3 x^-3)

That's how far I got, I think it's wrong. If anyone could help me out, I would appreciate it. Thanks.

 

[NoMoreExams]

How did you come up with your y'

 

[rappa]

24xy = x^4 + 48

y = (x^4 + 48) / 24x

y = 1/24 x^3 + 2/x or 2x^-1

dy/dx = 3/24 x^2 - 2x^-2

 

[NoMoreExams]

Now show me how you do this:

then: square root(1 + (3/24 x^2 -2x^2)^2)

= square root(1 + (1/64 x^4 - 4x^-4))

[You probably mean 1 + (3/24 x^2 - 2/x^2)^2)]

 

[rappa]

the 1+ part isn't squared, just the inside bracket. I just left the square root part, but I only squared the inside.

 

[NoMoreExams]

My post does not indicate that it is, does it?

My first point was that you wrote 2x^2 whereas you meant to write 2/x^2 or 2x^(-2)

My second point is show me how (3/24 x^2 - 2/x^2)^2 = 1/64 x^4 - 4x^(-4)

 

[rappa]

is it: 1/64 x^4 - 1/2 + 4/x^4 ?

 

[NoMoreExams]

That looks better

 

[rappa]

ok so now I have:

(2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)

 

[NoMoreExams]

I have no idea how you got that, you have to show more steps, for example next step would be to show what happens when you add 1 to what you had and take sqrt

 

[rappa]

We can't add the one, and we can't just take the square root either. Have to just integrate it straight from the square root and plug in the numbers to get the length.

This is what I did for the integration

(1/64 x^4 - 1/2 + 4/x^4)^(1/2)

= (2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)

 

[NoMoreExams]

...why not?

The steps are:

1) Take the derivative
2) Square it
3) add 1 to it
4) take square root of that
5) integrate

 

[rappa]

Ok here goes, I hope that works...

(1 + (1/64 x^4 - 1/2 + 4/x^4)

65/64 x^4 + 1/2 + 5/x^4

so now square root of that:

root(65/65) x^2 + root (1/2) + root (5) / x^2


that looks bad, that's why I thought to do it the other way:

(1/64 x^4 - 1/2 + 4/x^4)^(1/2)

integrates to:

(2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)

 

[NoMoreExams]

You should know from algeba that sqrt(a+b) is NOT sqrt(a) + sqrt(b), stop thinking that.

To work with what you have, find the common denominator and then use the property that sqrt(a/b) DOES equal sqrt(a)/sqrt(b)