Calculating Arc Length for a Polynomial Function on a Given Interval

[Alex G]

Homework Statement

F(x) = (4/5)*x^(5/4) on the interval of [0,4]
Find the Arc Length

Homework Equations

Arc Length = Integral (sqrt (1 + [f(x)']^2)) dx

The Attempt at a Solution

F'(x) = x^(1/4)

Integral from [0,4] of Sqrt (1 + x^(1/2)) dx

I'm not sure where to go with this integral, I'm sure it's a substitution, however I've been at this all day and I have no idea what.

 

[Dick]

Try the substitution u^2=1+x^(1/2).

 

[Alex G]

Okay, when I take the derivative of the u^2 = 1 + x^(1/2) to sub for dx I start to get lost. Would I do
u= sqrt(1 + x^(1/2)) and then take the derivative for du = dx part?

 

[Dick]

2*u*du=(1/2)*x^(-1/2)*dx. So 4*u*x^(1/2)*du=dx. Substitute u^2-1 for x^(1/2).

 

[Alex G]

Herp'a Derp'a ... thank you! Been doing so much of these Power and Taylor Series, I forgot the beginning stuff :(