CO-2 Multiple Integrals

Introduction to Multiple Integrals

• Definite Integral Extension: In single-variable calculus, the definite integral of a continuous function $f(x)$ over an interval $[a, b]$ is defined as a limit of Riemann sums. Multiple integrals extend this foundational concept to functions of two variables $f(x, y)$ over planar regions and functions of three variables $f(x, y, z)$ over spatial regions.
• Geometric Analogy: Just as a single definite integral can represent signed area in two dimensions, a double integral can represent signed volume in three dimensions.
• Evaluation Method: Double integrals are evaluated by applying the Fundamental Theorem of Calculus twice sequentially through iterated integration with respect to each variable.
• Coordinate Transformations: Complex double integrals can often be simplified using polar coordinates or general coordinate transformations, while triple integrals can be simplified using cylindrical or spherical coordinates.
• Applications: Multiple integrals are utilized to calculate area, volume, average values of multivariable functions, mass, moments, and centers of mass.

Double Integrals over Rectangular Regions

• Rectangular Region Definition: Consider a closed and bounded rectangular region $R$ in the $xy$-plane defined by:     $R: a \le x \le b, \quad c \le y \le d$
• Partitioning the Region: The region $R$ is subdivided into $n$ small rectangular subrectangles by a network of lines parallel to the coordinate axes.
• Area of Subrectangles: A small rectangular element with width $\Delta x_k$ and height $\Delta y_k$ has an area of:     $\Delta A_k = \Delta x_k \Delta y_k$
• Riemann Sum Construction: Choosing an arbitrary sample point $(x_k, y_k)$ inside the $k$-th subrectangle, the Riemann sum $S_n$ over $R$ is:     $S_n = \sum_{k=1}^{n} f(x_k, y_k) \Delta A_k$
• Norm of the Partition: The norm of a partition $P$, denoted as $||P||$, is