Triple Integrals Again the triple integral always exists if f is continuous We can choose the sample point to be any point in the subbox but if we choose it to be the point xi yj zk we get a simpler looking expression for the triple integral 10 Triple Integrals Just as for double integrals the practical method for evaluating triple integrals is to express them as iterated integrals as follows The iterated integral on the right side of Fubini s Theorem means that we integrate first with respect to x keeping y and z fixed then we integrate with respect to y keeping z fixed and finally we integrate with respect to z 11 Triple Integrals There are five other possible orders in which we can integrate all of which give the same value For instance if we integrate with respect to y then z and then x we have 12 Example 1 Evaluate the triple integral B xyz2 dV where B is the rectangular box given by B x y z 0 x 1 1 y 2 0 z 3 Solution We could use any of the six possible orders of integration If we choose to integrate with respect to x then y and then z we obtain 13 Example 1 Solution cont d 14 Triple Integrals Now we define the triple integral over a general bounded region E in three dimensional space a solid by much the same procedure that we used for double integrals We enclose E in a box B of the type given by Equation 1 Then we define F so that it agrees with f on E but is 0 for points in B that are outside E By definition This integral exists if f is continuous and the boundary of E is reasonably smooth 15 Triple Integrals The triple integral has essentially the same properties as the double integral We restrict our attention to continuous functions f and to certain simple types of regions 16 Triple Integrals 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 E x y z x y D u1 x y z u2 x y where D is the projection of E onto the xy plane as shown in Figure 2 A type 1 solid region Figure 2 17 Triple Integrals Notice that the upper boundary of the solid E is the surface with equation z u2 x y while the lower boundary is the surface z u1 x y By the same sort of argument it can be shown that if E is a type 1 region given by Equation 5 then The meaning of the inner integral on the right side of Equation 6 is that x and y are held fixed and therefore u1 x y and u2 x y are regarded as constants while f x y z is integrated with respect to z 18 Triple Integrals In particular if the projection D of E onto the xy plane is a type I plane region as in Figure 3 A type 1 solid region where the projection D is a type I plane region Figure 3 19 Triple Integrals In particular if the projection D of E onto the xy plane is a type I plane region as in Figure 3 then E x y z a x b g1 x y g2 x u1 x y z u2 x y and Equation 6 becomes 20 Triple Integrals If on the other hand D is a type II plane region as in Figure 4 then E x y z c y d h1 y x h2 y u1 x y z u2 x y and Equation 6 becomes A type 1 solid region with a type II projection Figure 4 21 Triple Integrals A solid region E is of type 2 if it is of the form E x y z y z D u1 y z x u2 y z where this time D is the projection of E onto the yz plane see Figure 7 The back surface is x u1 y z the front surface is x u2 y z and we have A type 2 region Figure 7 22 Triple Integrals Finally a type 3 region is of the form E x y z x z D u1 x z y u2 x z where D is the projection of E onto the xz plane y u1 x z is the left surface and y u2 x z is the right surface see Figure 8 A type 3 region Figure 8 23 Triple Integrals For this type of region we have In each of Equations 10 and 11 there may be two possible expressions for the integral depending on whether D is a type I or type II plane region and corresponding to Equations 7 and 8 24 Applications of Triple Integrals 25 Applications of Triple Integrals The triple integral E f x y z dV can be interpreted in different ways in different physical situations depending on the physical interpretations of x y z and f x y z Let s begin with the special case where f x y z 1 for all points in E Then the triple integral does represent the volume of E 26 Applications of Triple Integrals For example you can see this in the case of a type 1 region by putting f x y z 1 in Formula 6 and we know this represents the volume that lies between the surfaces z u1 x y and z u2 x y 27 Example 5 Use a triple integral to find the volume of the tetrahedron T bounded by the planes x 2y z 2 x 2y x 0 and z 0 Solution The tetrahedron T and its projection D onto the xy plane are shown in Figures 14 and 15 Figure 14 Figure 15 28 Example 5 Solution cont d The lower boundary of T is the plane z 0 and the upper boundary is the plane x 2y z 2 that is z 2 x 2y Therefore we have Notice that it is not necessary to use triple integrals to compute volumes They simply give an alternative method for setting up the calculation 29 Applications of Triple Integrals All the applications of double integrals can be immediately extended to triple integrals For example if the density function of a solid object that occupies the region E is x y z in units of mass per unit volume at any given point x y z then its mass is and its moments about the three coordinate planes are 30 Applications of Triple Integrals The center of mass is located at the point where If the density is constant the center of mass of the solid is called the centroid of E The moments of inertia about the three coordinate axes are 31 15 8 Triple Integrals in Cylindrical Coordinates Copyright Cengage Learning All rights reserved 32 Triple Integrals in Cylindrical Coordinates In plane geometry the polar coordinate system is used to give a convenient …
View Full Document
Unlocking...