Under what conditions does the Riemann sum approximation of volume converge to the double integral value?

Conditions

• The maximum diameter of any sub-region in the partition tends to zero ($||T|| \to 0$).
• The function $f(x,y)$ is defined on the region $D$.
• For the specific geometric intuition presented, $f(x,y)$ is assumed to be continuous and non-negative.

Reasoning, step by step

1. Start with a coarse partition of region $D$.
2. Calculate the Riemann sum using sample points in each sub-region.
3. Refine the partition by reducing the size of the grid cells.
4. Monitor the change in the sum as the mesh size decreases.
5. Identify the limit of the sum as the mesh size approaches zero.
6. Verify that this limit equals the double integral $\iint_D f(x,y)d\sigma$.

Example

The script states: 'As the grid partitions become increasingly dense, this stepped approximation model converges infinitely close to the actual smooth surface.' The card adds: 'For an integrable function, prism sums approach the double integral as the partition mesh tends to zero. Increasing the number alone is insufficient.'

Common misconceptions

• Believing that simply increasing the number of subdivisions $n$ is sufficient for convergence without ensuring the size of each subdivision shrinks to zero (e.g., adding more subdivisions only in one direction while others remain large).
• Assuming convergence holds for any bounded function, ignoring potential discontinuities that might prevent integrability (though Riemann integrability handles some discontinuities, the geometric intuition relies on smoother behavior).

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Explore next

• What is the precise definition of the 'norm of the partition' $||T||$?
• Does the location of the sample points $(\xi_i, \eta_i)$ matter for the final limit?