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To approximate the volume, partition the square $[-2,2] \times [-2,2]$ into four equal subsquares, each with side length 2 and area $\Delta A_k = 2^2 = 4$. Choose a sample point in each subsquare (e.g., the bottom-left corner).

Conditions: The base region is $[-2,2] \times [-2,2]$.; The partition consists of four equal subsquares.; Sample points are chosen according to a specific rule (e.g., bottom-left corners).

To approximate the volume, partition the square $[-2,2] \times [-2,2]$ into four equal subsquares, each with side length 2 and area $\Delta A_k = 2^2 = 4$. Choose a sample point in each subsquare (e.g., the bottom-left corner).

Conditions: The base region is $[-2,2] \times [-2,2]$.; The partition consists of four equal subsquares.; Sample points are chosen according to a specific rule (e.g., bottom-left corners).

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 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.

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.