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MA 242.003 . Day 43 – 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. . To quote your textbook: “ Just as we defined single integrals for functions of one variable.
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MA 242.003 • Day 43 – 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.
To quote your textbook: “Just as we defined single integrals for functions of one variable
To quote your textbook: “Just as we defined single integrals for functions of one variable
To quote your textbook: “Just as we defined single integrals for functions of one variable, and double integrals for functions of two variables
To quote your textbook: “Just as we defined single integrals for functions of one variable, and double integrals for functions of two variables
To quote your textbook: “Just as we defined single integrals for functions of one variable, and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.”
To quote your textbook: “Just as we defined single integrals for functions of one variable, and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.”
To quote your textbook: “Just as we defined single integrals for functions of one variable, and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.” Let’s first deal with the simple case where f is defined on a rectangular box
To quote your textbook: “Just as for double integrals the practical method for evaluating triple integrals is to express them as iterated integrals as follows.”
To quote your textbook: “Just as for double integrals the practical method for evaluating triple integrals is to express them as iterated integrals as follows.”
To quote your textbook: “Just as for double integrals the practical method for evaluating triple integrals is to express them as iterated integrals as follows.”
To quote your textbook: “Just as for double integrals the practical method for evaluating triple integrals is to express them as iterated integrals as follows.”
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
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
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.
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
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 To evaluate we will concentrate on certain SIMPLE REGIONS
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
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
Using techniques similar to what was needed for double integrals one can show that We have reduced the problem to a double integral over the region D
This formula simplifies if the projection D of E onto the xy-plane is type I.
When the formula Specializes to
When the formula Specializes to
When the formula Specializes to
When the formula Specializes to
