I take AP Calculus AB at my high school and I think I have found a flaw in my textbook. In a section about the length of curves, there is this problem asking the reader to find a curve based on a given integral and the part b of the problem asks how many curves are possible. The back of the book says that there is only one curve possible, but I think I have found another.

The problem is

Find a curve through the point (1,1) whose length integral is

so following the formula for the length of a curve, (dy/dx)^2=(1/4x) so (dy/dx) = + or -

. based on those dy/dx's, y is equal to either the square root of x plus a constant or the negative square root of x plus a constant, depending on whether you use the positive dy/dx or the negative one. After solving for constants I find two solutions that work:

and

but my text book only thinks the first one is an acceptable answer apparently. can someone help me out?