I think your limits are ok. You have a notational error here:
But as that gets fixed later on, that won't explain what's going on. I think part of the issue is that you've determined that There's no a priori reason to assume that. I would probably have done where is the radius of the base, to account for differently "squashed" cones. Be that as it may, I think your formula is actually correct, because in your case, since you have that and hence
This formula is correct if
Try going through the integration again with and see what that gets you.
Something else I would try: don't use both for the variable of integration and for the radius of the base of the cone. It's confusing.