13.5 Triple Integrals

# 13.5 Triple Integrals -  ...

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Unformatted text preview:   13.5  Triple Integrals in Rectangular Coordinates  Single Integral    fx a b     Domain:                                       Area:      Double Integral    Domain:                                            Area:                                                       Volume:                                             Mass:      Triple Integral    Domain:    Volume:      Mass:      ∫∫∫ f ( x, y, z)               E dV =         Volume:    ∫∫∫ (1) dV = ∫∫ =∫ ∫ E             S ( ∫ 1 dz ) ∫ Mass:  If density is  δ         dzdxdy   ( x, y, z) , then mass =  ∫∫∫ E   2 Ex.  Evaluate                                                          3y 8− x 2 − y 2 0 x 2 + 3y 2 ∫∫∫ 0 dzdxdy   Ex.  Let E = solid bounded by the planes x = 0, x = 2,  y = 0,   z = 0, and y + z = 1.  Find the volume of the solid.                                                            Ex.  Set up integrals to find the volume of the region formed by  x = 4y2 + 4z2 and the plane x = 4.                                                             Ex.  Find the volume of the region bounded by y + z = 1, y = x2,  and z = 0.                                                            Average value of a function F over a region D =     1 F dV         volume of D ∫∫∫ D   Ex. Find the average value of  f ( x, y, z) = x  on the region  bounded by y + z = 1, y = x2, and z = 0.                                                  Do: 1.  Set up an integral to find the volume of the region  bounded by the coordinate planes and 2x + 3y + 6z = 12.    2. Set up an integral to find the volume of the region cut  from the cylinder x2 + y2 = 4, the plane z = 0 and the plane   x + z = 3.                                                      Ex.  Sketch the solid whose volume is give by   1 1− x 2− 2z 00 0 dydzdx .  1 xy ∫∫ ∫               Ex.  Write 5 other iterated integrals that are equal to the given  integral                                  ∫∫∫ 000 f ( x, y, z) dzdydx .    Ex.  Evaluate the integral by changing the order of integration  in an appropriate way.  11 ∫∫ ∫ 0                                                 3 ln 3 z0 πe 2 x sin πy 2 dxdydz 2   y   ...
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## This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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