## Line Integral for a 3-D shape

So here is the question:
Compute the area of the portion of the parabolic cylinder of equation y=0.5x^2 that lies in the first octant and is bounded by the planes y=0, y=2, z=0 and by the elliptical cone z^2=x^2+4y^2

After hours of messing around with papers... and doing arts and crafts, I've figured out which "area" it is taking about.

Basically I'm supposed to get the area of the line y=0.5x^2 along z from 0 to rad 20 and the equation for the elliptical cone.

I think... I'm supposed to integrate the line integral of y=0.5x^2 along the bound set up by the elliptical cone... But I am completely lost as to how to do this.

I also did figure out that if I parametrize this... I would get x=t y=0.5t^2 so... if I plug this into my elliptical cone I get something like z= sqrt(t^4+t^2)