MA 242.003

• Day 44 – March 14, 2013• Section 12.7: Triple Integrals

GOAL: To integrate a function f(x,y,z) over a bounded 3-dimensional solid region in space.

Step 1: Subdivide the box into subboxes.

Generalization to bounded regions (solids) E in 3-space:

Generalization to bounded regions (solids) E in 3-space:

1. To integrate f(x,y,z) over E we enclose E in a box B

2. Then define F(x,y,z) to agree with f(x,y,z) on E, but is 0 for points of B outside E.

3. Then Fubini’s theorem applies, and we define

Definition: A solid region E is said to be of type 1 if it lies between the graphs of two continuous functions of x and y, that is

Using techniques similar to what was needed for double integrals one can show that

When

the formula

Specializes to

When

the formula

Specializes to

(continuation of problem 11)

Definition: A solid region E is said to be of type 2 if it lies between the graphs of two continuous functions of y and z, that is

Definition: A solid region E is said to be of type 2 if it lies between the graphs of two continuous functions of y and z, that is

(continuation of problem 17)

Definition: A solid region E is said to be of type 3 if it lies between the graphs of two continuous functions of x and z, that is

Definition: A solid region E is said to be of type 3 if it lies between the graphs of two continuous functions of x and z, that is

(continuation of problem 18)

An Application of Triple Integration

The volume of the solid occupying the 3-dimensional region E is

An Application of Triple Integration

The volume of the solid occupying the 3-dimensional region E is

An Application of Triple Integration

The volume of the 3-dimensional region E is

The area of the region D is

(continuation of problem 20)

#33

(continuation of problem 33)

(continuation of problem 33)

(see maple worksheet)

(continuation of problem 38)

(continuation of problem 43)

(continuation of problem )

(continuation of problem )