2D Range Sum Query

To query sum over rectangle $(r_1, c_1)$ to $(r_2, c_2)$, use inclusion-exclusion:

$\text{sum} = Q(r_2, c_2) - Q(r_1-1, c_2) - Q(r_2, c_1-1) + Q(r_1-1, c_1-1)$

where $Q(x, y)$ is the prefix sum to $(x, y)$.

def rangeSum(r1, c1, r2, c2):
    return (query2D(r2, c2)
          - query2D(r1-1, c2)
          - query2D(r2, c1-1)
          + query2D(r1-1, c1-1))

You're applying the same inclusion-exclusion principle used for 2D prefix sums, but now with $O(\log n \log m)$ queries and updates instead of $O(1)$ queries with $O(nm)$ updates.