Arc lengh properties

Arc lengh properties

Postby Guest » Tue Nov 24, 2020 12:05 am

Hey guys, i need some help with this excersice below.
Find the equation of the curve that passes through (0,1) using the following property:
-The area under the curve limited by the coordinate axes and the ordinate of any point is equal to the length of the curve corresponding to that region.
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Re: Arc lengh properties

Postby Guest » Thu Nov 26, 2020 11:53 am

The length of curve y= y(x) from x= a to x= b is given by [tex]\int_a^b \sqrt{1+ y'(x)^2} dx[/tex]. The area under that curve is given by [tex]\int_a^b y(x)dx[/tex]. So the condition that "The area under the curve limited by the coordinate axes and the ordinate of any point is equal to the length of the curve corresponding to that region" is that [tex]\int_0^x \sqrt{1+ y(t)^2} dt= \int_0^x y(t)dt[/tex].

So [tex]\sqrt{1+ y'(x)^2}= y(x)[/tex].

[tex]1+ y'(x)^2= y^2(x)[/tex].

[tex]y'(x)^2= y^2- 1[/tex].

[tex]y'(x)= \sqrt{y^2- 1}[/tex].

[tex]\frac{dy}{\sqrt{y^2- 1}}= dx[/tex].

[tex](y^2- 1)^{-1/2}= dx[/tex].
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