15 Multiple Integrals Copyright Cengage Learning All rights reserved Change of Variables in Triple Integrals 15 10 Copyright Cengage Learning All rights reserved Change of Variables in Triple Integrals Learning objective simplify triple integrals by a change of variables and then evaluate 3 Triple Integrals We have seen a formula for a change of variables for double integrals There is a similar change of variables formula for triple integrals Let T be a transformation that maps a region S in uvw space onto a region R in xyz space by means of the equations x g u v w y h u v w z k u v w 4 Triple Integrals The Jacobian of T is the following 3 3 determinant Under hypotheses similar to those in Theorem 9 we have the following formula for triple integrals 5 Example Use Formula 13 to derive the formula for triple integration in spherical coordinates Solution Here the change of variables is given by x sin cos y sin sin z cos We compute the Jacobian as follows 6 Example Solution continued cos 2 sin cos sin2 2 sin cos cos2 sin sin2 cos2 sin2 sin2 2 sin cos2 2 sin sin2 2 sin Since 0 we have sin 0 7 Example Solution continued Therefore and Formula 13 gives f x y z dV f sin cos sin sin cos 2 sin d d d 8
View Full Document
Unlocking...