Arc Length Problem

[speedychaos4]

How to evaluate the length of arc of y=x^4/4 from on [0,a]? thanks.

[origin415]

The formula for arc length is

$$\int_0^a \sqrt{(1 + (\frac{dy}{dx})^2} dx$$

Sorry I edited it to simplify as you were posting!

[speedychaos4]

First you want to parametrize x and y, so
x(t) = t
y(t) = t^4/4

then the formula for arc length is

$$\int_0^a \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$$

I'm ok with this formula...but I'm stuck with the integral...so i want to ask for help, thanks.

[origin415]

Well you can't actually do that integral, so don't try it (http://www.wolframalpha.com/input/?i=In ... %29%2Cx%29). What are the multiple choice answers? Maybe you can estimate the answer if the answers are not too close to one another.

If a is small, than $$\sqrt{1+x^6}$$ is close to 1, if a is large, its close to $$x^3$$.