Why is the finite Riemann sum written with $\approx$ instead of $=$?

Conditions

• The sum involves a finite number $n$ of subrectangles.
• The surface $f(x,y)$ is curved (not a flat plane).

Reasoning, step by step

1. Recognize that a finite partition uses flat-topped boxes to approximate a curved surface.
2. Observe that local errors occur where the box height differs from the surface height.
3. Understand that summing these imperfect boxes yields an approximate total volume.
4. Conclude that equality ($=$) is reserved for the limiting process where the error vanishes.

Example

The lecturer explicitly states: 'this is not exactly the answer, but if I'm going to do an approximation...' before introducing the sum. The on-screen formula for Step 3 uses $\approx$: Volume $\approx \sum_{k=1}^n f(x_k,y_k) \Delta x_k \Delta y_k$. Step 4 then uses $=$ with the limit: Volume $= \lim_{\|P\|\to 0} \sum_{k=1}^n f(x_k,y_k) \Delta x_k \Delta y_k$.

Common misconceptions

• Believing that a sufficiently large finite sum equals the exact volume.
• Thinking the approximation symbol is just a stylistic choice rather than a mathematical necessity.
• Confusing the finite sum with the definite integral.

Watch the explanation

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