Lecture Notes: Triple Integrals (Section 16.3)

Overview

• Previous Learning: Integration of functions started from single variable to double integrals for two variables.
• Current Focus: Triple integrals for functions of three variables, leading to the concept of hypervolume in four-space.

Key Concepts

Integrating Functions

• Single Variable Integration: Over an interval on the x-axis resulting in length.
• Double Integral: Over a region in the xy-plane resulting in volume.
• Triple Integral: Over a 3D region for functions of three variables.

Triple Integrals

• Order of Integration: dx, dy, dz can be in any order with appropriate bounds.
• Regions:
  • 1D (Single variable)
  • 2D (Double integral)
  • 3D (Triple integral)

Conceptualizing in Four-Space

• Function of Three Variables: f(x, y, z) lives in four-space.
• Result: Hypervolume, the 4D equivalent of volume.
• Visualization Challenge: Difficult to visualize 4D directly.

Setting Up Triple Integrals

Box in Three-Space

• Define 3D regions analogous to rectangular prisms.
• Bounds:
  • x: a to b
  • y: c to d
  • z: p to q

Calculating Volume

• Dividing Region: Into smaller regions with equal volumes.
• Delta Values:
  • Δx = (b-a)/n
  • Δy = (d-c)/m
  • Δz = (q-p)/l

Integral Setup

• Sum of hypervolumes: f(x_ijk, y_ijk, z_ijk) ΔV.
• Limit Process: As Δx, Δy, Δz → 0, integral of f(x, y, z) dV over the region.

Example Calculation

Problem

• Integrate f(x, y, z) = x y z^2 over a defined box.
• Bounds:
  • x: 0 to 1
  • y: -1 to 2
  • z: 0 to 3

Steps

• Choose order dz dy dx.
• Evaluate innermost to outermost integrals.
• Result: Hypervolume equals 27/4.

Sketching Regions

Example: Function of Three Variables

• Function: x + y in four-space.
• Region W: Defined by x/y/z bounds and equations.

Visualizing

• Determine intercepts for planes.
• Sketching intersections in 3D space.

Advanced Examples

Describing Regions

• Identify integration regions based on given bounds.
• Interpreting Bounds and Equations: Convert bounds to familiar geometric shapes (e.g., cylinders, spheres).

Example Sketch

• Problem: Sketch region W from integral bounds.
• Visualize: Determine region characteristics and visualize in 3D space based on bounds and order of integration.

Concluding Remarks

• Focus: Understanding setup and integration over 3D regions with triple integrals.
• Key Skill: Interpreting bounds and orders for correct integral setup and result interpretation.