How is the volume under a surface over a rectangular region defined using a double Riemann sum?

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.

Reasoning, step by step

1. Partition the rectangular region $[a,b] \times [c,d]$ into $n$ subrectangles.
2. Calculate the area of the $k$-th subrectangle as $\Delta A_k = \Delta x_k \Delta y_k$.
3. Choose a sample point $(x_k,y_k)$ in each subrectangle.
4. Evaluate the function $f(x_k,y_k)$ to find the height of the box over that subrectangle.
5. Sum the volumes of all boxes to get an approximation: $\sum_{k=1}^n f(x_k,y_k) \Delta x_k \Delta y_k$.
6. Take the limit of the sum as the partition norm $\|P\| \to 0$ to define the exact volume.

Example

The video displays the four steps on a chalkboard: 1) Partition the region $[a,b] \times [c,d]$ into little rectangles $\Delta A_k = \Delta x_k \Delta y_k$. 2) Choose a point $(x_k,y_k)$ in each rectangle. 3) Volume $\approx \sum_{k=1}^n f(x_k,y_k) \Delta x_k \Delta y_k$. 4) Volume $= \lim_{\|P\|\to 0} \sum_{k=1}^n f(x_k,y_k) \Delta x_k \Delta y_k$.

Common misconceptions

• Thinking the finite sum equals the exact volume; the video emphasizes it is only an approximation ($\approx$).
• Believing $\|P\| \to 0$ means the number of rectangles goes to zero; it actually means the largest rectangle's area goes to zero, causing the number of rectangles to go to infinity.
• Assuming the sample point $(x_k,y_k)$ must be a specific corner; the video states any point inside the rectangle works in the limit.

Watch the explanation

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