The Double Integral: Definition and Properties
From area under a curve to volume under a surface — the Riemann construction upgraded one dimension
1. The Volume Problem — Area Under a Curve, One Dimension Up
1.1 Recall: The Single-Variable Riemann Integral
In single-variable calculus you built the definite integral $\int_a^b f(x)\,dx$ by:
- Partitioning $[a,b]$ into subintervals of width $\Delta x$.
- Choosing a sample point $x_i^*$ in each subinterval.
- Forming the Riemann sum $\displaystyle S_n = \sum_{i=1}^{n} f(x_i^*)\,\Delta x$.
- Taking the limit as $n \to \infty$ (all widths $\to 0$).
Each term $f(x_i^*)\,\Delta x$ is the area of a thin rectangle. The limit gives the exact area under the curve $y = f(x)$ above $[a,b]$.
Callback: This is the Riemann construction you mastered in single-variable calculus. Everything below is the same story told in two dimensions.
1.2 The Upgrade: Volume Under a Surface
Now let $f(x,y)$ be a function of two variables defined over a plane region $R$ (think of a rectangle $R = [a,b] \times [c,d]$ in the $xy$-plane). The graph $z = f(x,y)$ is a surface hovering above $R$. We want the volume of the solid squeezed between $R$ and the surface — exactly as we wanted the area between the $x$-axis and a curve, but one dimension higher.
1.3 The Construction — Executed Honestly
| Step | What You Do |
|---|---|
| Partition | Divide $R$ into small cells (subrectangles) of area $\Delta A_k$. If $R = [a,b]\times[c,d]$, cut $[a,b]$ into $m$ pieces of width $\Delta x$ and $[c,d]$ into $n$ pieces of width $\Delta y$, giving $m \times n$ cells each of area $\Delta A = \Delta x \cdot \Delta y$. |
| Sample | In each cell $R_{ij}$, choose a sample point $(x_{ij}^*,\, y_{ij}^*)$. |
| Sum | Build the double Riemann sum: $$S_{m,n} = \sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{ij}^*,\, y_{ij}^*)\,\Delta A$$Each term is the volume of a thin rectangular column (base $\Delta A$, height $f(x_{ij}^*,y_{ij}^*)$). |
| Refine & Limit | Let $m,n \to \infty$ (all cell diameters $\to 0$). The limit, if it exists and is the same for every choice of partition and sample points, is the double integral. |