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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.

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 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.

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.