Solving a Parabolic Bowl Oscillation Problem

[jderm]

Homework Statement

Consider a particle moving back and forth on a frictionless parabolic bowl, y = ax2, where a = 1.460 m^-1
If the particle is released from rest at the point on the
bowl at b = 0.43 m, find the period of the oscillations.

I have an equation for velocity(as a function of x). What i was thinking is that i could integrate this from -b to +b, and call this value 'v(x)*d', (it having units of m²/s.
since v=d/t, v(x)*d=d²/t.
thus t=d²/'v(x)*d'.
making the period, T=2*(d²/'v(x)*d'),
where d is the arc length of y(x), from -b to +b.

I realize this may be the least elegant and very possibly "physically incorrect", but it actually gave me a period very close to that obtained by the 'small angle approximation' (however, incorrect)

Can anybody comment on this method?

 

[jderm]

i guess the crux of my argument is whether the area under the v(x) curve can be used in this manner, comments?

 

[RoyalCat]

The acceleration and velocity are not at all constant, so the method is, mathematically speaking, completely wrong.