Why must a point $(x_k,y_k)$ be chosen inside the subrectangle?

Q: Why must a point $(x_k,y_k)$ be chosen inside the 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. The chosen point $(x_k,y_k)$ provides this specific input, and $f(x_k,y_k)$ gives the corresponding height for the box's volume calculation.

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. The chosen point

(x_k,y_k)

provides this specific input, and

f(x_k,y_k)

gives the corresponding height for the box's volume calculation.

Conditions

The domain is partitioned into subrectangles.
A function $f(x,y)$ defines the surface height.

Reasoning, step by step

Recognize that a subrectangle contains infinitely many points.
Understand that the function $f(x,y)$ varies across the subrectangle.
Select a single sample point $(x_k,y_k)$ inside the subrectangle.
Evaluate $f(x_k,y_k)$ to get a specific height value.
Use this height to calculate the volume of the box over that subrectangle.

Example

The speaker explains: 'because the subrectangle is a whole little region, he needs to choose a specific point to plug into the function... different people can make different choices as to where that point is going to be.' The 2D diagram shows a yellow point labeled $(x_k,y_k)$ inside the red rectangle.

Common misconceptions

Thinking the height of the box is automatically determined by the subrectangle's boundaries.
Believing there is only one correct point to choose inside the subrectangle.
Assuming the function is constant over the entire subrectangle.

Watch the explanation

YouTubeDefining Double Integration with Riemann Sums | Volume under a Surface

2:29 – 2:59Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.

Why must a point (xk,yk)(x_k,y_k)(xk​,yk​) be chosen inside the subrectangle?