Change of Variables in Double Integrals The determinant that arises in this calculation is called the Jacobian of the transformation and is given a special notation With this notation we can use Equation 6 to give an approximation to the area A of R where the Jacobian is evaluated at u0 v0 1 Change of Variables in Double Integrals The foregoing argument suggests that the following theorem is true Theorem 9 says that we change from an integral in x and y to an integral in u and v by expressing x and y in terms of u and v and writing 2 Change of Variables in Double Integrals Notice the similarity between Theorem 9 and the one dimensional formula in Equation 2 Instead of the derivative dx du we have the absolute value of the Jacobian that is x y u v As a first illustration of Theorem 9 we show that the formula for integration in polar coordinates is just a special case 3 Change of Variables in Double Integrals Here the transformation T from the r plane to the xy plane is given by x g r r cos y h r r sin and the geometry of the transformation is shown in Figure 7 T maps an ordinary rectangle in the r plane to a polar rectangle in the xy plane The polar coordinate transformation Figure 7 4 Change of Variables in Double Integrals The Jacobian of T is Thus Theorem 9 gives 5 15 7 Triple Integrals Copyright Cengage Learning All rights reserved 6 Triple Integrals We have 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 simplest case where f is defined on a rectangular box B x y z a x b c y d r z s The first step is to divide B into sub boxes We do this by dividing the interval a b into l subintervals xi 1 xi of equal width x dividing c d into m subintervals of width y and dividing r s into n subintervals of width z 7 Triple Integrals The planes through the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub boxes Bijk xi 1 xi yj 1 yj zk 1 zk which are shown in Figure 1 Each sub box has volume V x y z Figure 1 8 Triple Integrals Then we form the triple Riemann sum where the sample point is in Bijk By analogy with the definition of a double integral we define the triple integral as the limit of the triple Riemann sums in 9 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 …
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