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 December 6th, 2010, 11:30 AM #1 Newbie   Joined: Dec 2010 Posts: 11 Thanks: 0 Length of a curve I have a task to calculate the length of the curve. Thatīs right? Does anyone know outcome? Formula: derivative exponentiation
 December 6th, 2010, 11:47 AM #2 Senior Member   Joined: Nov 2010 Posts: 288 Thanks: 1 Re: Length of a curve (dy/dx)^2=(x-1)^2/4x (not (x-1)^2/2x as you wrote) now 1+ (dy/dx)^2= (4x+x^2-2x+1)/4x= (x^2+2x+1)/4x= (x+1)^2/4x now if you take the root you get (x+1)/2root(x)= x/2root(x) +1/(2root(x)) = (1/2)*root(x) +1/(2root(x)) now the integration is easy = (1/3)x^(3/2)+root(x)
 December 6th, 2010, 02:16 PM #3 Senior Member   Joined: Nov 2010 Posts: 288 Thanks: 1 Re: Length of a curve thanks amillion for editting i am pretty grateful
 December 11th, 2010, 02:25 AM #4 Newbie   Joined: Dec 2010 Posts: 11 Thanks: 0 Re: Length of a curve thank you very much
December 11th, 2010, 06:17 AM   #5
Math Team

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Re: Length of a curve

Hello, snopy!

Quote:
 $\text{I have to calculate the length of the curve: }\;y \;=\;\frac{1}{3}(x\,-\,3)\cdot\sqrt{x}$

I would do it like this . . .

$\text{We have: }\;y \;=\;\frac{1}{3}\left(x^{\frac{3}{2}} \,-\, 3x^{\frac{1}{2}}\right)$

$\text{Differentiate: }\;y' \;=\;\frac{1}{3}\left(\frac{3}{2}x^{\frac{1}{2}} \,-\, \frac{3}{2}x^{-\frac{1}{2}}\right) \;=\;\frac{1}{2}\left(x^{\frac{1}{2}} \,-\, x^{-\frac{1}{2}}\right)$

$\text{Square: }\;(y')^2 \;=\;\frac{1}{4}\left(x^{\frac{1}{2}}\,-\,x^{-\frac{1}{2}}\right)^2 \;=\;\frac{1}{4}\left(x\,-\,2\,+\,x^{-1}\right)$

$\text{Add 1: }\;1 + (y')^2 \;=\;1\,+\,\frac{1}{4}\left(x\,-\,2\,+\,x^{-1}\right) \;=\;\frac{1}{4}\left(x\,+\,2\,+\,x^{-1}\right) \;=\;\frac{1}{4}\left(x^{\frac{1}{2}}\,+\,x^{-\frac{1}{2}}\right)^2$

$\text{Square root: }\;\sqrt{1\,+\,(y')^2} \;=\;\sqrt{\frac{1}{4}\left(x^{\frac{1}{2}}\,+\,x^ {-\frac{1}{2}}\right)^2} \;=\;\frac{1}{2}\left(x^{\frac{1}{2}}\,+\,x^{-\frac{1}{2}}\right)$

$\text{Then: }\;L \;=\;\frac{1}{2}\int^{\;\;\;\;b}_a\left(x^{\frac{1 }{2}}\,+\,x^{-\frac{1}{2}}\right)\,dx$

 December 13th, 2010, 07:21 AM #6 Newbie   Joined: Dec 2010 Posts: 11 Thanks: 0 Re: Length of a curve Hi, soroban thank you so much

 Tags curve, length