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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 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.

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.