Approximating Area Under a Curve with Rectangles

Goal: determine the exact area that lies
- below the curve y = f(x)
- above the x–axis
- to the right of the vertical line x = a
- to the left of the vertical line x = b
Standard mathematical notation for this desired area: the definite integral \int_a^b f(x)\,dx
Referred to throughout as “the area problem.”

When the region is a simple geometric figure (e.g.
- rectangles,
- triangles,
- semicircles),
  basic area formulas are sufficient.
Strategy for complex curves: decompose the complicated region into many simple regions whose areas we already know how to compute.
Chosen shape for decomposition in this lecture: rectangles (simplest two-dimensional figure with an easily computed area formula).

Begin with a single rectangle spanning the interval [a,b].
- Area formula used: \text{Area}=\text{width}\times\text{height}.
- Width: b-a.
- Height: some representative function value on [a,b].
- Result: rough estimate; large portions may be omitted (gaps) or included in excess (overhang).
Motivation to improve estimate by increasing rectangle count.
For two rectangles:
- Interval [a,b] is split in half; two heights chosen, one per sub-interval.
- Smaller gaps/overhangs ⇒ noticeably better approximation.
For six rectangles (illustrated example):
- Each rectangle tailored to a shorter sub-interval so its top edge tracks the curve more faithfully.
- Leads to “less gap, less overhang.”
General observation:
- As the number of rectangles n increases and each rectangle becomes narrower, the collective shape visually merges with the curve itself.
- With sufficiently many narrow rectangles (e.g., tens, hundreds, or thousands), the eye may not detect a difference between the broken rectangular outline and the true curved boundary.

Visual intuition:
- 1 rectangle → coarse.
- 2 rectangles → noticeably better.
- 6 rectangles → clear fidelity improvement.
- 20–50 rectangles → discrepancies become tiny pixels on a normal display.
Conceptual takeaway: thinner rectangles + greater quantity ⇒ approximation error shrinks.
No predetermined “correct” n; rather, we can push n "as large as we like" until the approximation satisfies any practical tolerance.

Replace informal language (“a bunch of rectangles”) with precise mathematical symbols.
Sum of areas of n rectangles:
\sum_{i=1}^{n} \big(\text{width of rectangle } i \times \text{height of rectangle } i\big)
Translating the verbal plan:
\inta^b f(x)\,dx \;\approx\; \sum{i=1}^{n} \text{Area}i where \text{Area}i stands for the i–th rectangle’s area.
Ultimate aspiration: turn “≈” into “=” by letting n grow without bound:
\inta^b f(x)\,dx \;=\; \lim{n\to\infty} \sum{i=1}^{n} \text{Area}i
This limit of rectangle areas is the foundational concept behind Riemann sums and, by extension, the rigorous definition of the definite integral.
Familiar mathematical tools now at our disposal:
- Summation notation (\Sigma) for “add them all up.”
- Limit notation for “let n approach infinity.”
- These tools place the area problem on “firmer ground”—we move from pictures to precise, manipulable symbols.

The area under a curve—once seemingly beyond elementary geometry—can be approached systematically by slicing the region into rectangles whose individual areas are trivial to compute.
Accuracy is directly linked to the number and thinness of rectangles; the theoretical ideal uses infinitely many rectangles of infinitesimal width.
The formal statement \inta^b f(x)\,dx = \lim{n\to\infty}\sum{i=1}^{n} \text{Area}i will evolve into the definition of the definite integral via Riemann sums in subsequent lessons.
Moving forward, we’ll refine how to choose rectangle heights (left-end, right-end, midpoint, etc.) and explore how the limiting process yields exact values for a wide variety of functions.