Chapter 8: Applications of Triple Integration
Section 8.5: Changing Variables in a Triple Integral
If the equations x=xu,v,w,y=yu,v,w,z=zu,v,w define an appropriate change of coordinates whereby a region R is mapped to a region R′, then the following equality of two iterated triple-integrals is valid,
∫∫∫Rfx,y,z dv = ∫∫∫R′fxu,v,w,yu,v,z,zu,v,w ∂x,y,z∂u,v,w dv′
where dv is one of six possible orderings of the differentials dx,dy,dz, and dv′ is one of six possible orderings of the differentials du,dv,dw. The integral on the right also contains the absolute value of the Jacobian
Each column in the Jacobian matrix is differentiated with respect to a single variable, whereas the rows are gradients of the functions xu,v,w,yu,v,w,zu,v,w.
In addition, the Jacobian ∂x,y,z∂u,v,w is the reciprocal of the Jacobian ∂u,v,w∂x,y,z, the Jacobian of the inverse transformation, because the corresponding Jacobian matrices are multiplicative inverses of each other.
Find the total mass in R, the interior of the ellipsoid x/22−y/32−z/42=1, if its density is δx,y,z=1−x/22−y/32−z/42.
Calculate the volume of R, the region bounded by the hyperbolic cylinders x y=1, x y=5, x z=4, x z=15, y z=10, yz=20.
Using cylindrical coordinates, calculate the volume of R, the region bounded by the paraboloids z=4 x2+y2 and z=9x2+y2 and the planes z=2 and z=3. Then, recalculate the volume by making the change of variables x=u/vcosw, y=u/vsinw, z=u2.
(Versions of these three examples can be found as exercises in texts such as Calculus 6e (Early Transcendentals, Matrix Version) by Edwards & Penney, Prentice Hall, 2002; and Schaum's Outlines - Advanced Calculus by Spiegel, McGraw-Hill, 1996.)
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