Rectangular Approximation and Integration by First Principles

Goal: Compute the exact area under a curve y=f(x) on the closed interval [a,b].
Historical context of last lesson: used basic geometry (rectangles) as a conceptual bridge to calculus.
Key idea: If we can sum the areas of enough rectangles, their total area should converge to the true area beneath the curve.

Choose rectangles because their area formula is trivial: \text{Area}=\text{width}\times\text{height}.
Intuition: Increasing the number of rectangles (and shrinking their width) yields a better approximation.
Limiting case: As the number of rectangles n\to\infty, the approximation ideally becomes exact.

All rectangles share the same width.
- Partition [a,b] into n equal sub-intervals.
- Common width (denoted \Delta x): \Delta x=\frac{b-a}{n}.
Equal widths simplify both the algebra and limit process.

Label left end of the interval: x_0=a.
Label right end: x_n=b.
General formula for the right endpoint of the i^{\text{th}} rectangle (or the i^{\text{th}} partition point): x_i=a+i\,\Delta x
- Verification examples:
- x_0=a+0\cdot\Delta x=a (matches a).
- x_n=a+n\,\Delta x=a+(b-a)=b (matches b).
This creates an evenly spaced grid along the x-axis.

Width is fixed ((\Delta x)), but height must track the curve.
One common choice (used in this lecture): Right-endpoint evaluation.
- Height of the i^{\text{th}} rectangle: f(x_i).
- Geometric meaning: The rectangle’s upper-right corner lies on the curve.
Alternatives (not elaborated here, but conceptually relevant): left-endpoint, midpoint, or maximum/minimum sample inside each sub-interval.

Area of the i^{\text{th}} rectangle:
Ai = f(xi)\,\Delta x
Total approximate area with n rectangles (Riemann sum):
Sn = \sum{i=1}^{n} f(x_i)\,\Delta x
Take the limit as the number of rectangles grows without bound: \inta^b f(x)\,dx = \lim{n\to\infty} \sum{i=1}^{n} f(xi)\,\Delta x
- This expresses the definite integral purely from first principles (no antiderivatives used).

If the above limit exists and equals some finite value, we say:
- "f(x) is integrable on [a,b]."
Analogy to differentiation:
- Just as existence of a derivative at a point implies "differentiable," existence of this integral implies "integrable."
Practical implication: The method works only if the function behaves "well enough" for the limit to settle (e.g.
boundedness, limited discontinuities).

This rectangle‐limit formulation is often called:
- Integration by first principles
- Riemann integration from definition
Importance: Provides the rigorous foundation before introducing shortcuts (antiderivatives, Fundamental Theorem of Calculus, numerical integration, etc.).

Equal-width rectangles transform a qualitative idea ("lots of little boxes") into a quantitative method.
Summation notation and limits furnish the bridge from algebra to calculus.
The construction mirrors tangent-slope limits in differentiation, underscoring the symmetry between the two central operations of calculus.
Once the limit is shown to exist, the definite integral gives the exact area, completing the "area problem" by first principles.
Future lectures typically exploit antiderivatives to avoid computing explicit limits, but this foundational method validates those shortcuts.