Understanding double integrals

The convergence occurs when the partition of the base region $D$ becomes arbitrarily fine, specifically when the maximum mesh size (norm of the partition, denoted $||T||$ or $||\mathcal P||$) approaches zero. Additionally, the function $f(x,y)$ must be integrable over $D$; in the context of the video's geometric visualization, continuity and non-negativity are cited as justifying factors for the specific example shown.

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.

The definition is justified when the height function $f(x,y)$ is continuous and nonnegative over the base region D. This ensures that the surface is smooth enough and lies above the base, allowing the Riemann sum limit to represent a physical volume.

Conditions: The function $f(x,y)$ is continuous.; The function $f(x,y)$ is nonnegative ($f(x,y) \ge 0$).; The base region D is a closed planar region.

The volume is approximated by uniformly partitioning the base region into small rectangular grids and constructing vertical rectangular prisms on each cell. The height of each prism corresponds to the surface value at a sample point, and the sum of these prism volumes forms a rough approximation of the true volume.

Conditions: The base region is a closed planar region D.; The surface is smooth and upward-bulging.; The grid partitions are uniform.

The volume is approximated by partitioning the base region $D$ into small rectangular grids. On each grid cell, a vertical rectangular prism is constructed with height equal to the function value $f(\xi_i, \eta_i)$ at a sample point within that cell.

Conditions: The solid is bounded above by a smooth surface $z=f(x,y)$ and below by a planar region $D$.; The base region $D$ is uniformly partitioned into tiny rectangular grids.; $\Delta\sigma_i$ represents the area of the $i$-th sub-region.; $(\xi_i, \eta_i)$ is a sample point chosen within the $i$-th sub-region.

The infinitesimal base area $\Delta\sigma$ represents the area of a small grid cell on the base region D, while $f(\xi_i, \zeta_i)$ represents the height of the curved surface above that specific sample point. Their product approximates the volume of a single rectangular prism.

Conditions: The base region is subdivided into small grid cells.; A sample point $(\xi_i, \zeta_i)$ is chosen within each cell.; The surface height is evaluated at the sample point.

The formal definition is given by the limit of the sum of the products of the function height and the infinitesimal base area, as the maximum mesh size of the partition approaches zero. Mathematically, it is expressed as $V = \iint_D f(x,y)d\sigma = \lim_{||T||\to0} \sum_{i=1}^n f(\xi_i, \zeta_i)\Delta\sigma_i$, where $f(\xi_i, \zeta_i)$ is the height at a sample point and $\Delta\sigma_i$ is the area element.

Conditions: The function $f(x,y)$ represents the height of the surface above the region $D$.; The limit is taken as the norm of the partition $||T||$ (maximum diameter of sub-regions) goes to zero.; The sum extends over all $n$ sub-regions of the partition.

The stepped approximation converges because as the grid partitions become increasingly dense, the maximum mesh size tends to zero. This causes the discrete sum of prism volumes to approach the continuous integral, infinitely close to the actual smooth surface volume.

Conditions: The grid partitions become increasingly dense.; The maximum mesh size tends to zero.; The surface is smooth and continuous.