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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:.

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ma 242 003
MA 242.003
  • Day 44 – March 14, 2013
  • Section 12.7: Triple Integrals
slide2

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

slide9

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

slide10

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

slide13

When

the formula

Specializes to

slide15

When

the formula

Specializes to

slide18

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

slide19

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

slide24

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

slide25

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

slide30

An Application of Triple Integration

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

slide31

An Application of Triple Integration

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

slide32

An Application of Triple Integration

The volume of the 3-dimensional region E is

The area of the region D is

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