Section 15.1

Double Integrals

Extending the definite integral to find volume under a surface.

1

Introduction

In single-variable calculus, represents the area under the curve .

In multivariable calculus, the Double Integral represents the volume under the surface over a region .

2

Geometric Interpretation

Interactive: Volume Under Surface

3

Properties

  • Linearity: .
  • Constant Multiple: .
  • Additivity: If , then .
  • Average Value: The average value of on is .
4

Worked Examples

Example 1: Volume of a Box

Calculate where .

1. Geometric Method:

Since is constant, this is just a rectangular box.

.

.

.

Example 2: Riemann Sum Approximation

Estimate on using 4 squares (midpoints).

Squares are . Area .

Midpoints:

Sum:

.

Example 3: Average Value

If and is a circle of radius 5, find the average value of f.

1. Find Area:

.

2. Use Formula:

.

Answer: .

5

Practice Quiz

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