Finding the length of the curve (where it gets undefined at t=0)

[hangainlover]

Homework Statement

Find the length of the curve

f(x)= x^(1/3)+ x^(2/3), 0≤x≤2

Homework Equations

L= integral of square root of (1+ (dy/dx)^2 (dx) ) or (1+ (dx/dy)^2 (dy) )

The Attempt at a Solution

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?

 

[mighty2000]

Thank you for your question. The method you have described is correct for finding the length of a curve between two points using the formula L= ∫√(1+(dy/dx)^2)dx (or L= ∫√(1+(dx/dy)^2)dy). However, in this case, you are asked to find the length of the curve between two specific values of x, not between two specific points. To solve this problem, you can use the formula L= ∫√(1+(dy/dx)^2)dx and evaluate it between the given values of x, 0 and 2. This will give you the length of the curve between those two values of x. I hope this helps. Best of luck with your problem!