Question

Find the length of the curve. You may use your calculator.
f(x)=x^(1/3)+x^(2/3) [0,2]
I understand that the function needs to be written as x in terms of y because there's a vertical tangent at x=0, but I don't understand how to go about the problem other than that.

Answer 1

The vertical slope does not matter, once you go through the hoops. The answer is shown here:

http://www.wolframalpha.com/input/?i=plot+x%5E(1%2F3)%2Bx%5E(2%2F3)+for+x%3D0..2

The problem I have is evaluating that integral! I tried something like

u = 1+2∛x
du = 2/3 x^(-2/3)
x = (u-1)^3/8

and the integral then becomes

∫1/2 √(9(u-1)^4/8 + u^2) du

which is no easier to handle. Looks like this is really a calculator problem.

Answer 2

analyze how Wolfram does it.
https://www.wolframalpha.com/input/?i=arc+length+of+%3Dx%5E(1%2F3)%2Bx%5E(2%2F3)++from+x%3D0+to+x%3D2