Changing Order of Integration for Multiple Integrals Click here for a double integral.

In a Nut Shell: The evaluation of double or triple integrals over a region, R, may

be complicated due to the complexity of the integrand, due to the order of integration,

or both. The strategy is the same for both double and triple integrals but the triple

integrals are generally harder since you are dealing with intersecting surfaces that

are harder to visualize rather than intersecting curves.

Strategy: Identify the surfaces (or curves) that bound the region of integration by drawing

the projections of the surfaces on various planes (or the curves). In so doing you will identify

the limits of integration in each plane or the points of intersection of the curves. Then pick

an order of integration that simplifies evaluation of the integral.

Example: Change the order of integration for the integral below from dy dz dx

to dy dx dz. See the figure below.

x = 1 z = 1-x² y = 1 - x

I  =  ∫  ∫ ∫  f(x,y,z) dy dz dx

x = 0 z = 0 y = 0

The first step is to visualize, R, the region of integration and the intersecting surfaces

by using limits of integration to plot the region. i.e. y = 1 ˗ x is a plane in the z-direction.

z = 1 ˗ x² is a surface in the y˗direction.

Click here to continue this example.

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