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What is the geometric role of the sample point (xk,yk)(x_k,y_k) in determining the box height?

The sample point (xk,yk)(x_k,y_k) serves as the specific input location where the surface function f(x,y)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)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 (xk,yk)(x_k,y_k) is chosen inside that subrectangle.

Reasoning, step by step

  1. Identify the subrectangle in the domain partition.
  2. Choose a sample point (xk,yk)(x_k,y_k) within it.
  3. Evaluate the function f(xk,yk)f(x_k,y_k) to find the surface height at that point.
  4. Construct a vertical box with base ΔAk\Delta A_k and height f(xk,yk)f(x_k,y_k).
  5. Calculate the box volume as f(xk,yk)ΔxkΔykf(x_k,y_k) \Delta x_k \Delta y_k.

Example

The video shows a 3D surface with a vertical box rising from a base rectangle. The height of the box is explicitly labeled f(xk,yk)f(x_k,y_k), connecting the 2D sample point to the 3D geometric volume element. The lecturer says: 'taking f of that point gives the height of the box.'

Common misconceptions

  • Thinking the sample point determines the base area of the box.
  • Believing the height is determined by the corners of the subrectangle rather than the chosen interior point.
  • Confusing the sample point with the centroid of the subrectangle (though it can be the centroid, it doesn't have to be).

Watch the explanation

3:17 – 4:01Watch 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.