## 11.3 How to Find Limits

Suppose your variables are u,v and w

Find the variable you wish to integrate over first: suppose it is u;

Fix all the other variables, v and w at arbitrary values v' and w', and compute the values of u that are in the desired volume for these values of v' and w'. Normally these values define one or more intervals for each value of v'and w': integrate u over these intervals.

Then pick the next variable to integrate; say v; fix w at w'and find the values of v for which there is any value u' of u such that (u', v, w') is in your volume; these are normally intervals, and you integrate v over these intervals. Finally you integrate w over all values for which there is any pair (u', v') which lies in your volume.

Another way to say the same thing: integrate over all values of u for fixed v' and w'; then collapse (or project) the region onto the v w plane; fix w and integrate over v; collapse (or project) the region onto the w axis, and integrate over w.

Example