Throwing Stuff Down A Mineshaft

I think the easier way to picture the integration inside the shell is the following:Think of a slender cone emanating from you in opposite directions. Where this cone intercepts the sphere surrounding you is the mass pulling you in that direction. But the amount of mass intercepted and the gravitational pull of that mass vary as distance squared. Gravity is diminished by distance squared. The mass intercepted is increased by distance squared. So the effect of the pull from the window of this cone is constant and exactly countered by the effect of the intercepted mass of the cone directed oppositely to the first. In the limit as the steradian measure of this cone is diminished the results become exactly accurate. This nullifies the effects of the cone not being perpendicular to the shell at all summation orientations.

Actually the angles of interception I think are the same, as any straight line is a chord in a circle and a great circle can be drawn on the sphere through any two point of interception. Sorry, a little rusty.