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26-Triple Integrals - Triple Integrals John E Gilbert...

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Triple Integrals John E. Gilbert, Heather Van Ligten, and Benni Goetz Once we’ve seen the pattern of how to go from integrals of functions of one variable to double integrals of functions of two variables, then studying triple integrals E f ( x, y, z ) dxdydz of functions f ( x, y, z ) of three variables is a natural step, and a very important one in applications. Now the domain of integration is a solid E in 3-space, and when f > 0 the integral is interpreted as the volume of the solid in 4-space below the graph of f and above E . Examples show how to write the triple integral as a repeated integral: Example 1: express the triple integral I 1 = E f ( x, y, z ) dxdydz as a repeated integral when E is the solid shown to the right above the rectangle D = [ a, b ] × [ c, d ] in the xy -plane and below the graph of z = 6 - x . Think of D as the base of E and the graph of z = 6 - x above D as the top of E . Now fix a point P = ( x, y ) in D , and let z go vertically along the red line from P up to the black dot at the top.
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