You do not have to calculate integrals given in the Wikipedia article for a simple airplane simulation. Those integrals describe the moment of inertia (or to be more precise moment of inertia tensor) and have been calculated for many common shapes. So if you make some assumptions like uniform mass distribution and use simple plane geometry, there is no need to integrate at all.

The simplest way I would have start with is to describe plane as a set of point masses, each corresponding to different part of the plane. This way the calculations should be very easy. Momentum of inertia is an additive quantity, so all you have to do is calculate it for each point and then add it together.

The formula for momentum of inertia of a mass point is also simple:
mass multiplied by square of the distance to the rotation axis.

Here are some links
Moment of inertia
List of moments of inertia for some common shapes.