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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.

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.