Double Integrals over Rectangles

Definition

The double integral of $f(x, y)$ over a rectangular region $R = [a, b] \times [c, d]$ is defined as the limit of Riemann sums as the norm of the partition approaches zero: $\iint_R f(x, y) dA = \lim_{n \to \infty} \sum_{k=1}^n f(x_k, y_k) \Delta A_k$

• How to read: “The double integral over the region R of f of x, y with respect to A equals the limit as n approaches infinity of the sum from k equals one to n of f of x k, y k times delta A k.”
• Meaning: Riemann sum definition—partition the rectangle, weight f at each patch by its area, and refine until the sum converges to total accumulation (e.g., volume under a surface).

Why It Matters

You cannot understand a 3D volume by only looking at 2D slices in isolation. Double integrals over rectangles matter because they provide the first rigorous “bridge” from 2D area to 3D accumulation. They are the foundation for “volumetric thinking,” allowing us to calculate the total pressure on a dam or the total weight of a building slab where the density varies at every single point on the surface.

Core Concepts

• Partition and Subrectangles: Dividing $R$ into small subrectangles $\Delta A_k$.
• Riemann Sum: The sum $\sum f(x_k, y_k) \Delta A_k$ approximating the volume under a surface.
• Existence: If $f(x, y)$ is continuous on $R$, the double integral exists.

Connected Concepts

• Area by Double Integration and Average Value by Double Integration
• Double Integrals over General Regions
• Double Integrals in Polar Form