1.Find the volume of the ellipsoidal solid given by
Solution:
Because x,y, and z play similar roles in the equation, the order of integration is probably immaterial, and you arbitrarily choose
.
Moreover, you can simplify the calculation by considering only the portion of
the ellipsoid lying in the first octant. From the order
,
you first determine the bounds for z:
Then, the boundaries for x any are:
Thus, you solve the following integral:
Use integration tables in the next step:
2. Evaluate
Solution:
Note that after one integration in the given order, you would encounter the
integral
which is not an elementary function. To avoid this problem, change the order
of integration to
,
so that y is the outer variable. The solid region Q is given by:
From this we can notice that the projection of Q in the xy plane yields the
bounds:
Therefore:
