# triple integral iteration exercise

Instructions to accompany this one page PDF printout handed out in class:

We will analyze this as in this lecture example: tripleintexample.pdf
[or this one:  tripleintexample2.pdf with maple worksheet  tripleintegraltexample2.mw].

In case my instructions are not perfectly clear on the instructions written on this exercise PDF, here is what I expect and will explain in class. If you miss class, read on.

Look at the 3-dimensional diagram at the top right representing the innermost of the triple integrals, with three highlighted strips on the left side aligned with the three coordinate axes in the diagram. In the diagram itself, for each variable doing the innermmost integration, put in a typical line segment cross-section parallel to the axis of the variable of that innermost integration with bullet point endpoints labeled by the starting and stopping values of that variable which increases along the line segment (put an arrow head in the middle for this increasing direction). This requires your imagination in estimating where in each surface you should start and stop in order to convey that the endpoints actually lie on the starting and finishing surfaces. When you finish the solid region should have 3 such typical labeled cross-sections representing the 3 possible innermost integrations.

For each of the three 2-dimensional projection diagrams of the solid region into the coordinate planes at the bottom of the page (representing the outer double integral of the triple integral over that solid region in space shown in the 3-dimensional diagram in the top right), place both a typical horizontal line segment cross-section and a typical vertical line segment cross-section with bullet endpoints labeled by the equations which set the starting and stopping values of the coordinate which increases as you move along that line segment (bottom to top, left to right), including an arrowhead in the middle of that line segment indicating that increasing direction of the variable of integration. These tell us the limits for the innermost integral of the outer double integral, while the  ranges of the outermost variable needed to sweep those cross-sections across the plane region must be read off from the axis tickmarks. In the online PDF there is highlighting in those diagrams where you should put the two linear cross-sections, and below there are two highlighted regions to input the starting and stopping expressions for the third variable associated with the innermost of the triple integration. Refer to the completed 3-dimensional diagram of the solid region in the upper right to write down these expressions.

Using your diagrams at the bottom of the page, using either the space below them at the bottom of the page or on a new sheet of paper, write down the six corresponding iterated integrals with a unit integrand.
Finally set the integrand equal to 1 to evaluate all six integrals using Maple to get their values. They should all same value: the volume of the solid region is 8/15. This gives you some confidence that perhaps your iterations are all correct (but might give the same result by coincidence).