多重积分的数值逼近

The definition doubles as an algorithm

A worked midpoint computation

Regions beyond rectangles and higher dimensions

Reporting honestly

The midpoint rule samples each cell at which point?

Refining the grid by a factor of two in each direction reduced an estimate's error by roughly a factor of four. What law is on display?

An integral over a complicated region resists every substitution and order change. What is the sound response?

Estimate $\iint_{[0,2] \times [0,2]} x y\, dA$ with one midpoint cell, giving the estimate as a plain number.

A midpoint estimate carries error $0.08$ on a given grid. After refining by two in each direction, the expected error is $0.02$. Give the expected error after one more identical refinement, as a decimal like 0.005.