Find the lenght of the curve
f(x)= x^(1/3)+ x^(2/3), 0≤x≤2
L= integral of square root of (1+ (dy/dx)^2 (dx) ) or (1+ (dx/dy)^2 (dy) )
The problem is that i cannot take the inverse function of that and differentiate it.
For example, if you are given y=x^(1/3) and asked to find the curve between (-8,-2) and (8, 2),
1. switch x and y (reverse the x y coordinates so (-2,-8), (2,8) and you have x=y^(1/3)
2. cube both sides x^3=y
3, differentiate it : 3x^2=dy/dx
4, plug that into the formula of L and evaluate it from -2 to 2.
I cannot do the same for the given function f(x)= x^(1/3)+ x^(2/3), 0≤x≤2
Because, after switching x and y, x=y^(1/3)+y^(2/3) +y^(2/3), even if you differentiate it, you cannot isolate y and define dy/dx only in terms of x.
What should i do?