Classical mechanics - Energy balance confussion

[JuanC97]

Hi everyone.
I'm confused with the balance of energy in this situation (I'm comparing the mechanical energy, initially, at rest, at a height = h, and then, at the end, at height = 0). It doesn't seem to be fine, maybe I missed something.
I'd appreciate some help - Everything is in the .jpg below.

There are 2 relevant equations:
1. The work done by a nonconservative force is equal to the change in mechanical energy. (Wnc = Delta E)
2. W = Force (dot) distance.
Also, take into account that h / sin(theta) is equal to the hypotenuse of the inclined plane.

 

[Doc Al]

It's a bit difficult to follow. And since you put your work in a jpg instead of typing it here, it is hard to comment.

One thing: You seem to mix up the height h with the distance along the surface.
And another: You may have mixed up some signs.

 

[JuanC97]

You seem to mix up the height h with the distance along the surface

Hmmm, right. y(t) gives me the displacement along the surface (not the height).
Give me some minutes and I'll have the equations fixed.

* EDIT: Doc Al, you were right. I rewrote the equations by hand and everything fits well now, thanks.
I just had to change the limits in one integral at the beginning of the deductions.