Double Integrals over Rectangular Regions

Introduction to Double Integrals

Double integrals extend the concept of integration to functions of two variables, allowing us to compute areas, volumes, and other quantities over a region in the plane. When the region is a rectangle, the process is particularly systematic and forms the foundation for more general cases.

• Double Integral: The double integral of a function f(x, y) over a rectangular region R = [a, b] \times [c, d] is denoted as .

• Applications: Used to find volumes under surfaces, average values, and other physical quantities.

Approximating Area and Volume: Riemann Sums

To approximate the area under a curve or the volume under a surface, we divide the region into small rectangles and sum the contributions from each.

• Partitioning the Region: Divide [a, b] into n intervals and [c, d] into m intervals, forming mn rectangles.

• Sample Points: In each rectangle, select a sample point (x_i, y_j).

• Riemann Sum for Area:

• Riemann Sum for Volume:

• Limits: As the number of rectangles increases (and their size decreases), the Riemann sum approaches the exact value of the double integral.

Example: Approximating Volume

• Given z = f(x, y) over R = [0, 2] \times [0, 3], divide [0, 2] into 2 intervals and [0, 3] into 3 intervals, forming 6 rectangles.

• Sample points: (0,0), (1,0), (0,1), (1,1), (0,2), (1,2)

• Compute at each point and sum:

Evaluating Double Integrals

For a continuous function f(x, y) over a rectangle R = [a, b] \times [c, d], the double integral can be computed as an iterated integral.

• Iterated Integral: or

• Order of Integration: The order (dx then dy, or dy then dx) can be chosen for convenience when the region is rectangular.

Example: Compute over

• First, integrate with respect to y:

• Inner integral:

• Outer integral:

Fubini's Theorem

Fubini's Theorem states that if f(x, y) is continuous on a rectangle R, then the double integral can be computed as an iterated integral in either order:

Average Value of a Function over a Region

The average value of a function f(x, y) over a region R is given by dividing the double integral by the area of the region.

• Formula:

• For a rectangle R = [a, b] \times [c, d],

Example: Average Value

• Given f(x, y) = x - 3y over R = [0, 2] \times [0, 1]

• Compute and divide by area

• Result:

Summary Table: Key Concepts in Double Integrals over Rectangular Regions

Additional info: Some steps and explanations were expanded for clarity and completeness, including the explicit calculation of Riemann sums and the use of Fubini's Theorem.