What is the geometric role of the infinitesimal base area $\Delta A_k$ in the volume element?

Conditions

• The domain is partitioned into subrectangles.
• $\Delta x_k$ and $\Delta y_k$ are the side lengths of the $k$-th subrectangle.

Reasoning, step by step

1. Identify the $k$-th subrectangle in the domain partition.
2. Calculate its area as the product of its side lengths: $\Delta A_k = \Delta x_k \Delta y_k$.
3. Recognize this area as the base of the vertical box standing over it.
4. Multiply this base area by the box height $f(x_k,y_k)$ to get the box volume.
5. Sum these box volumes to approximate the total volume under the surface.

Example

The video isolates one red rectangle in the 2D domain and labels its sides $\Delta x_k$ and $\Delta y_k$. It then shows a 3D box rising from this base, with height $f(x_k,y_k)$. The formula $\Delta A_k = \Delta x_k \Delta y_k$ is explicitly written in Step 1 of the definition.

Common misconceptions

• Thinking $\Delta A_k$ is the volume of the box itself.
• Confusing the base area with the total area of the region $[a,b] \times [c,d]$.
• Believing the height is determined by the base area rather than the function value at the sample point.

Watch the explanation

Explore next

• How is the volume under a surface over a rectangular region defined using a double Riemann sum?
• Why must a point $(x_k,y_k)$ be chosen inside the subrectangle?
• What is the geometric role of the sample point $(x_k,y_k)$ in determining the box height?