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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

(continuation of problem 11) integrals one can show that

Definition: integrals one can show thatA 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: integrals one can show thatA 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) integrals one can show that

Definition: integrals one can show thatA 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: integrals one can show thatA 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) integrals one can show that

An Application of Triple Integration integrals one can show that

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

An Application of Triple Integration integrals one can show that

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

An Application of Triple Integration integrals one can show that

The volume of the 3-dimensional region E is

The area of the region D is

(continuation of problem 20) integrals one can show that

#33 integrals one can show that

(continuation of problem 33) integrals one can show that

(continuation of problem 33) integrals one can show that

(see maple worksheet) integrals one can show that

(continuation of problem 38) integrals one can show that

(continuation of problem 43) integrals one can show that

(continuation of problem ) integrals one can show that

(continuation of problem ) integrals one can show that

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