Why does the exact choice of sample point $(x_k,y_k)$ not affect the final volume?

Conditions

• Applies in the limit as the partition norm $\|P\| \to 0$.
• Each $(x_k,y_k)$ must be chosen inside its corresponding subrectangle.

Reasoning, step by step

1. Acknowledge that for a finite partition, the choice of $(x_k,y_k)$ affects the sum's value.
2. Consider the limit process where subrectangles become infinitesimally small.
3. Recognize that as the subrectangle area approaches zero, the variation in function values within it also approaches zero.
4. Conclude that the limiting volume is independent of the specific sample point choice.

Example

The lecturer states at the beginning of the formal definition segment: 'where exactly you choose the $x_k$,$y_k$ inside of the rectangle is not going to actually matter' in the limit. The 2D diagram shows a yellow point labeled $(x_k,y_k)$ inside a red rectangle, illustrating that it can be anywhere inside.

Common misconceptions

• Believing that a specific rule (like always choosing the bottom-left corner) is required for the exact volume.
• Thinking that the finite sum's dependence on the sample point means the exact integral also depends on it.
• Assuming the sample point must be the centroid or a specific geometric center.

Watch the explanation

Explore next

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