Question

Find a curve with a positive derivative through the point (0,1) whose length integral is

Answer 1

\[L=\int\limits_{1}^{2}\sqrt{1+\frac{ 1 }{ y^{4} }}dy\]

Answer 2

b) how many such curves are there

Answer 3

y^3 = 3 (x - k)

prob infinite as k can be anything...if tat is what you mean....

Answer 4

The arc length of a curve \(C\) defined over an interval \([a,b]\) by a function \(f(y)\) is given by
\[L=\int_CdS=\int_a^b\sqrt{1+(f'(y))^2}\,dy\]
Right away, you can match some elements of this integral with the given integral. Clearly, \([a,b]\) is the interval \([1,2]\), and it must be that \(f'(y)=\pm\dfrac{1}{y^2}\), since \((f'(y))^2=\dfrac{1}{y^4}\).

Now you have a differential equation (a separable one, at that), plus an initial value \((0,1)\); that is, when \(y=0\), you have \(f(0)=1\).