Why is the area S first written as approximately equal to the rectangular sum here, rather than directly equal?

Q: Why is the area S first written as approximately equal to the rectangular sum here, rather than directly equal?

The area $S$ is written as approximately equal to the rectangular sum because, for any finite number of divisions $n$, the sum of the areas of the rectangles does not exactly match the area under the curve. There are small gaps or overlaps between the tops of the rectangles and the curve $f(x)$. Equality is only achieved in the limit as $n$ approaches infinity, where the width of each rectangle approaches zero and the approximation error vanishes.

The area

S

is written as approximately equal to the rectangular sum because, for any finite number of divisions

n

, the sum of the areas of the rectangles does not exactly match the area under the curve. There are small gaps or overlaps between the tops of the rectangles and the curve

f(x)

. Equality is only achieved in the limit as

n

approaches infinity, where the width of each rectangle approaches zero and the approximation error vanishes.

Conditions

$n$ is a finite positive integer.
The function $f(x)$ is not constant on the subintervals.
The limit $n \to \infty$ has not yet been taken.

Reasoning, step by step

Observe that the rectangles have a finite width $\frac{1}{n}$ .
Note that the top of each rectangle is a horizontal line segment, while the curve $f(x)$ may be curved.
Identify the small regions between the rectangle tops and the curve as approximation errors.
Conclude that the sum of rectangle areas is an approximation of the true area $S$ .
Recognize that exact equality requires taking the limit as $n \to \infty$ .

Example

The video explicitly writes $S \approx \frac{1}{n}\sum_{k=1}^{n} f\left(\frac{k}{n}\right)$ and states that when $n$ is fixed, the total area of the rectangles is usually not equal to the area under the curve, so the approximately equal sign is used instead of the equals sign.

Common misconceptions

Believing that a large finite $n$ (like $n=500$ ) makes the sum exactly equal to the area.
Confusing the visual appearance of continuity with mathematical equality.
Thinking that the approximation sign is just a notational convenience rather than reflecting a genuine mathematical difference for finite $n$ .

Watch the explanation

YouTubeDefinite integrals from animated rectangle sums

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Why is the area S first written as approximately equal to the rectangular sum here, rather than directly equal?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

Why is the area S first written as approximately equal to the rectangular sum here, rather than directly equal?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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