Section 15.5

Triple Integrals

Calculating Mass, Volume, and other properties of 3D Solids.

Introduction

Just as a single integral sums length to get area, and a double integral sums area to get volume, a Triple Integral sums volume elements .

If , the result is the .
If , the result is the .

Types of Regions

Type I (z-simple)

Integrate z first ("floor to ceiling").

Then D is the "shadow" of E in the xy-plane.

Visualizing a Tetrahedron

Interactive: Plane x+y+z=1

Worked Examples

Example 1: Mass of a Box

Find mass of box if density is .

1. Setup:

2. Integrate z:

3. Integrate y:

4. Integrate x:

Mass = 12.

Example 2: Volume of Tetrahedron

Find the volume of the solid bounded by and coordinate planes.

1. Define Limits:

z goes from 0 to .

Shadow in xy-plane (where z=0): .

y goes from 0 to .

x goes from 0 to 1.

Setup: .

2. Integrate z:

3. Integrate y:

4. Integrate x:

Example 3: Parabolic Wedge

Find volume bounded by cylinder and planes .

1. Visualize:

Floor is region bounded by and . Roof is .

2. Limits:

z: 0 to 4.

y: to 4.

x: -2 to 2 (intersection of ).

Setup: .

3. Solve:

Inner (z): .

Middle (y): .

Outer (x): (symmetry).