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The volume is defined by a four-step limiting process. First, partition the rectangular region $[a,b] \times [c,d]$ into small subrectangles with area $\Delta A_k = \Delta x_k \Delta y_k$.

Conditions: The base region must be a rectangle $[a,b] \times [c,d]$.; The function $f(x,y)$ defines the height of the surface.; The limit is taken as the largest subrectangle area approaches zero.

The volume is defined by a four-step limiting process. First, partition the rectangular region $[a,b] \times [c,d]$ into small subrectangles with area $\Delta A_k = \Delta x_k \Delta y_k$.

Conditions: The base region must be a rectangle $[a,b] \times [c,d]$.; The function $f(x,y)$ defines the height of the surface.; The limit is taken as the largest subrectangle area approaches zero.

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).

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.

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.

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.