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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 finite Riemann sum is written with $\approx$ because it represents an approximation of the volume, not the exact value. A finite number of boxes cannot perfectly match a curved surface; some boxes will protrude above the surface (overestimating) and others will fall below it (underestimating).

Conditions: The sum involves a finite number $n$ of subrectangles.; The surface $f(x,y)$ is curved (not a flat plane).

The finite Riemann sum is written with $\approx$ because it represents an approximation of the volume, not the exact value. A finite number of boxes cannot perfectly match a curved surface; some boxes will protrude above the surface (overestimating) and others will fall below it (underestimating).

Conditions: The sum involves a finite number $n$ of subrectangles.; The surface $f(x,y)$ is curved (not a flat plane).

In the limiting process where the partition becomes infinitely fine ($\|P\| \to 0$), the exact location of the sample point inside each subrectangle does not matter. While different choices of $(x_k,y_k)$ will yield different finite Riemann sums (approximations), the limit of these sums as the subrectangles shrink to zero size is independent of the specific interior point chosen.

Conditions: Applies in the limit as the partition norm $\|P\| \to 0$.; Each $(x_k,y_k)$ must be chosen inside its corresponding subrectangle.

In the limiting process where the partition becomes infinitely fine ($\|P\| \to 0$), the exact location of the sample point inside each subrectangle does not matter. While different choices of $(x_k,y_k)$ will yield different finite Riemann sums (approximations), the limit of these sums as the subrectangles shrink to zero size is independent of the specific interior point chosen.

Conditions: Applies in the limit as the partition norm $\|P\| \to 0$.; Each $(x_k,y_k)$ must be chosen inside its corresponding subrectangle.