3. Consider the pinched torus described in cylindrical coordinates by
$(r - 1)^2 + z^2 = 1$ (equivalent to $r^2 + z^2 = 2r$), which encloses a solid
region R of space shown in the figure, obtained by revolving a circle
about the $z$-axis, as shown in the $r$-$z$ diagrams below. Obviously
$\theta = 0..2\pi$ for this solid of revolution and is the outer variable of
integration of the triple integral. We analyze the inner double integral.
a) Express the triple integral $\iiint_R z^2 dV$ in cylindrical coordinates, and
evaluate using Maple.
Justify your limits for the $r$-$z$ inner double integral with the $r$-$z$ half plane diagram below shaded by equally spaced
vertical linear cross-sections representing the inner integral, labeling one typical one as usual, and fill-in the
starting and stopping values of the outer integration variable in that half plane.
b) Express the equation for this surface in spherical coordinates.
c) Express the triple integral $\iiint_R z^2 dV$ in spherical coordinates, justified by a similarly labeled $r$-$z$ half plane
diagram below representing the limits of integration for $\rho$ and $\phi$, and evaluate using Maple. It should agree with a).
d) Evaluate the volume $V$ of this region in spherical coordinates using Maple.
e) From c) and d) evaluate the average value of $z^2$ over this region.
cylindrical inner:
$z$
1
0
-1
outer range: $r =$
spherical inner:
$z$
1
0
-1
outer range: $\phi = $
