In single-variable calculus, the definite integral $\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$ captures the net area under a curve. As we step into the third dimension, we extend this logic to find the volume under a surface $z = f(x, y)$.
1. The Formal Definition
We define the double integral of a function $f$ over a closed rectangle $R = [a, b] \times [c, d]$ as the limit of a double Riemann sum:
$$\iint_R f(x, y) \, dA = \lim_{m, n \to \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*) \Delta A$$
where $\Delta A = \Delta x \Delta y$ is the area of a sub-rectangle $R_{ij}$, and $(x_{ij}^*, y_{ij}^*)$ is any sample point within $R_{ij}$.
1. Geometric Partitioning: Divide $R$ into $m \times n$ sub-rectangles $R_{ij}$ where $x_i = a + i\Delta x$ and $y_j = c + j\Delta y$.
2. Solid Approximation: For each $R_{ij}$, construct a column of height $f(x_{ij}^*, y_{ij}^*)$. The volume $V$ of the solid $S$ is approximated by $V \approx \sum \sum f(x_{ij}^*, y_{ij}^*) \Delta A$.
3. The Limit: As the grid becomes infinitely fine ($m, n \to \infty$), the approximation converges to the exact volume.
2. Average Value Theorem
Just as the 1D average height of a curve is $\frac{1}{b-a}\int f(x)dx$, the average value of a surface $z=f(x,y)$ over a region $R$ is:
$$f_{ave} = \frac{1}{A(R)} \iint_R f(x, y) \, dA$$
This $f_{ave}$ represents the height of a single rectangular box with base $R$ that would contain the same volume as the complex solid beneath the surface.