In physics, you often approximate a continuous weighting via a discrete one:
assume the mass is concentrated at its center of gravity.

For -torque-, this is exact,
  b/c you're integrating x, which is linear.
For -gravity- (& potentials),
  it is not, b/c you're integrating 1/x^2, which is not linear.
This leads you to -underestimate- the force.
2 ways to see this:
- convexity
  1/x^2 is convex up;
  thus secants lie above the curve;
  Given 2 points (of equal mass), the midpoint of the secant
  is the average force per unit mass,
  while the point on the curve below that is the force per unit mass
  at the midpoint, which is lower.
- you lose more for close points than for far points
  (which is a nice way of seeing asymptoting to zero and convexity):
  b/c the slope of 1/x^2 is decreasing,
  you understate the force of near points by moving them to the center
  -more- than you overstate the force of far points,
  so you understate overall.

"dipole correction" (no "monopole correction"?)

(BTW, someone should mention to little kids that potentials drop off as 1/x^2,
which is exactly how the surface area of an expanding sphere -grows-.
Coincidence? No.
  [likewise the connection with harmonic functions/fundamental solution;
   there's a lot of basic PDEs that really should be shared.]
Indeed, that's how you visualize little force-carrying particles.
...and you can thus imagine 2D or 4D physics,
with 1/x or 1/x^3 drop-off (and 1D physics with -log drop-off:
harmonic & all that))