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The infinitesimal base area $\Delta A_k = \Delta x_k \Delta y_k$ represents the footprint or base of a single representative rectangular box in the Riemann sum. It is the area of the $k$-th subrectangle in the domain partition.

Conditions: The domain is partitioned into subrectangles.; $\Delta x_k$ and $\Delta y_k$ are the side lengths of the $k$-th subrectangle.

The infinitesimal base area $\Delta A_k = \Delta x_k \Delta y_k$ represents the footprint or base of a single representative rectangular box in the Riemann sum. It is the area of the $k$-th subrectangle in the domain partition.

Conditions: The domain is partitioned into subrectangles.; $\Delta x_k$ and $\Delta y_k$ are the side lengths of the $k$-th subrectangle.

A point $(x_k,y_k)$ must be chosen because the subrectangle is a two-dimensional region, not a single input value. To determine the height of the representative box standing over that subrectangle, one must evaluate the function $f(x,y)$ at a specific location.

Conditions: The domain is partitioned into subrectangles.; A function $f(x,y)$ defines the surface height.

A point $(x_k,y_k)$ must be chosen because the subrectangle is a two-dimensional region, not a single input value. To determine the height of the representative box standing over that subrectangle, one must evaluate the function $f(x,y)$ at a specific location.

Conditions: The domain is partitioned into subrectangles.; A function $f(x,y)$ defines the surface height.

The notation $\|P\| \to 0$ means that the size of the largest subrectangle in the partition approaches zero. As the largest rectangle shrinks to zero area, all other rectangles in the partition must also shrink to zero.

Conditions: Applies to the limit step in the definition of volume by double Riemann sums.; $P$ represents the partition of the rectangular region.

The notation $\|P\| \to 0$ means that the size of the largest subrectangle in the partition approaches zero. As the largest rectangle shrinks to zero area, all other rectangles in the partition must also shrink to zero.

Conditions: Applies to the limit step in the definition of volume by double Riemann sums.; $P$ represents the partition of the rectangular region.

The sample point $(x_k,y_k)$ serves as the specific input location where the surface function $f(x,y)$ is evaluated to determine the vertical height of the representative rectangular box. Geometrically, it anchors the top of the box to the surface $z=f(x,y)$ directly above that point, allowing the box's volume to be calculated as base area times this specific height.

Conditions: A subrectangle has been defined in the domain.; A point $(x_k,y_k)$ is chosen inside that subrectangle.

The sample point $(x_k,y_k)$ serves as the specific input location where the surface function $f(x,y)$ is evaluated to determine the vertical height of the representative rectangular box. Geometrically, it anchors the top of the box to the surface $z=f(x,y)$ directly above that point, allowing the box's volume to be calculated as base area times this specific height.

Conditions: A subrectangle has been defined in the domain.; A point $(x_k,y_k)$ is chosen inside that subrectangle.