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The equation is: $$x=\frac{1}{8}y^4 + \frac{1}{4}y^{-2},\qquad 1\leq y\leq 2.$$

I have the formula. I'm not sure how to write it out but this is what it says:

Length is equal to the integral (with $b$ and $a$ for limits) of the square root of $1+(dy/dx)^2 dx$

So I first took the derivative of the equation and got $(1/2)y^3 - (1/2)y^{-3}$.

Now when I plug that back into the formula, I have to square it and I got $(1/4)y^6 - (1/4)y^{-6}$. I factored out a $1/4$ and turned the $y^{-6}$ into $1/y^6$. Just a mess at this point, that doesn't seem right. Can someone help me out with that? Haha I know it's ridiculous but my algebra skills are lacking.

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Did you notice that you do not have $y$ as a function of $x$, but rather $x$ as a function of $y$, so that your formula should use $\frac{dx}{dy}$, not $\frac{dy}{dx}$? –  Arturo Magidin May 24 '11 at 5:31
    
Note the artificiality of the problem, a feature it shares with the majority of arc length problems in calculus books. If you change the $1/8$ to $1/8.1$, you end up with something you cannot integrate in closed form. –  André Nicolas May 24 '11 at 6:12
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+1 for trying before asking –  Ross Millikan May 24 '11 at 12:20

1 Answer 1

up vote 5 down vote accepted

You squared incorrectly. The square of $a-b$ is not $a^2-b^2$. (Also, you should be using $\frac{dx}{dy}$, not $\frac{dy}{dx}$, because here your independent variable is $y$ and your dependent variable is $x$; your integral will be with respect to $y$, not with respect to $x$).

The square of $\displaystyle\frac{1}{2}y^3 - \frac{1}{2}y^{-3}$ is not just the difference of the squares (which is what you wrote). Rather, it is equal to: $$\left(\frac{1}{2}y^3 - \frac{1}{2}y^{-3}\right)^2 = \frac{1}{4}y^6 - 2\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)y^3y^{-3} + \frac{1}{4}y^{-6} = \frac{1}{4}y^6+\frac{1}{4}y^{-6} - \frac{1}{2}.$$ When you add $1$, you get $$\frac{1}{4}y^6 + \frac{1}{4}y^{-6} + \frac{1}{2} = \left(\frac{1}{2}y^3 + \frac{1}{2}y^{-3}\right)^2.$$

Remember: $(a+b)^2 = a^2 + 2ab+b^2$, and $(a-b)^2 = a^2-2ab+b^2$. The square of the sum is not the sum of the squares.

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Ah yes you're right about the dx/dy part, missed that. And wow I feel so stupid after realizing what I did on the squaring part. That's a mistake you should only make in middle school! Think I've been studying too much today. Thanks! –  Ryan May 24 '11 at 5:38

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