What is the Length of a Curve with Imaginary Values?

[blumfeld0]

hi. i am trying to find the length of the curve
Find the length of the curve
x = (3) *y^(4/3) - (3/32)*y^(2/3) from -64<_y<_27

so i do the usual

calculate( 1 + (dx/dy)^2 ) ^(1/2)

then do the integral from -64 to 27 with respect to y

the only thing is i get an imaginary answer!

someone could please shed some light on this

thank you

 

[HallsofIvy]

If you expect some one to tell you what you did wrong you will need to show exactly what you did! All I can say now is that you must have done some step of the calculation incorrectly. For this problem, [itex]1+ (\frac{dx}{dy})^2} is a perfect square: there is no square root in the integral so you couldn't possibly have got an imaginary answer!

 

[blumfeld0]

hi. well i did everything on mathematica

i took the derivative, squared it, added one and took the square root of the whole thing. then i integrated with respect to y from -27 to 64
i.e (1+ dx/dy^2)^1/2 from -27 to 64 with respect to y

can you please tell me what the perfect square is?

thanks

 

[BlackEagle]

then i integrated with respect to y from -27 to 64

Didn't you say you needed to integrate from -64 to 27?

 

[blumfeld0]

yeah sorry that's what i meant

-64 to 27

 

[HallsofIvy]

hi. well i did everything on mathematica

i took the derivative, squared it, added one and took the square root of the whole thing. then i integrated with respect to y from -27 to 64
i.e (1+ dx/dy^2)^1/2 from -27 to 64 with respect to y

can you please tell me what the perfect square is?

thanks

Well, if it isn't important enough for you to write out exactly what you did, it certainly isn't important enough for me to do that- it's not my problem!

 

[ObsessiveMathsFreak]

Two things.

Firstly this curve is a multivalued function, there are multiple answers unless you restrict yourself to a certain set. On my software at least, inputting (-1)^(1/3) evaluates to 1/2 + i*sqrt(3)/2, so it seem to be choosing one of the imaginary values here. If that's the case... something terrible will happen. In your case, the integral is imaginary.

Secondly, even if you fix the first problem, this curve has a discontinuity in its derivative at y=0. His will also result in horribleness if I remeber correctly.

Where did you get this curve? Perhaps the original parameterisation will be more forgiving.

 

[Office_Shredder]

Doing some algebraic manipulations by hand can usually radically alter the output of the computer. Even if it's just something simple like factoring f^1/3+f^-1/3 =f^1/3(1 + 1/f^2/3).

Although this is from using maple, not mathematica. It may not help you

 

[HallsofIvy]

If x = (3) y^(4/3) - (3/32)y^(2/3)
then x'= 4y^1/3- (1/16)y^-1/3
so x'²= 16y^2/3- (1/2)+ (1/256)y^-2/3
Then 1+ x'²= 16y^2/3+(1/2)+ (1/256)y^-2/3= (4y^1/3+ (1/16)y^-1/3)²
and its square root is
4y^1/3+ 1/16 y^-1/3

As others pointed out, that's an improper integral because it is not defined at y= 0 so you have to be careful about it.

On my software at least, inputting (-1)^(1/3) evaluates to 1/2 + i*sqrt(3)/2, so it seem to be choosing one of the imaginary values here.

That's the "principal third root of -1" but its odd that your software would give it as "the" root. In the real numbers, of course, the third root of -1 is -1.