Geometric Applications of the Definite Integral
A unified approach: Slice → Approximate → Sum → Integrate
1 — The Universal Slicing Recipe (Riemann-Sum Backbone)
Core Idea: Every formula in this topic is not a separate fact to memorize. Each one is the same four-step procedure applied to a different geometric or physical quantity.
The Four Steps
- Partition the domain (an interval $[a,b]$) into $n$ thin sub-intervals of width $\Delta x = \frac{b-a}{n}$.
- Approximate the quantity on each thin piece as a simple product:
$$\text{(something measurable)} \times \Delta x$$ For area this is $f(x_i^*)\,\Delta x$ (height × width). For volume it is $A(x_i^*)\,\Delta x$ (cross-sectional area × thickness). For mass it is $\rho(x_i^*)\,\Delta x$ (density × length), etc. - Sum over all pieces to get a Riemann sum: $$S_n = \sum_{i=1}^{n} (\text{quantity}_i)\,\Delta x$$
- Take the limit $n\to\infty$ (equivalently $\Delta x\to 0$). The Riemann sum becomes a definite integral: $$\lim_{n\to\infty} S_n = \int_a^b (\text{integrand})\,dx$$
Key insight: Once you master this recipe, you never need to memorize a formula list. You rebuild each formula on the spot by asking: “What simple shape approximates one thin slice?”
Interactive: Watch the Riemann Sum Converge
Interactive visualization — available in the full lesson
2 — Plane Areas in Cartesian Coordinates
2.1 Area Under a Single Curve
If $f(x)\ge 0$ on $[a,b]$, the area under the curve and above the $x$-axis is:
$$A = \int_a^b f(x)\,dx$$This is the direct output of the slicing recipe: each thin rectangle has height $f(x)$ and width $dx$.