You will have to provide values for the two radii:
Integrate[1, {x, y} \[Element] Circle[{0, 0}, {1, 2}]]
(*8 EllipticE[3/4]*)
and
NIntegrate[1, {x, y} \[Element] Circle[{0, 0}, {1, 2}]]
(*9.68845*)
Cheers,
Marco
PS: Note that this is the circumference not the "volume" as you suggest in your question. The area is easy:
Integrate[1, {x, y} \[Element] Disk[{0, 0}, {r1, r2}], Assumptions -> {r1 > 0, r2 > 0}]
(*\[Pi] r1 r2*)
and
NIntegrate[1, {x, y} \[Element] Disk[{0, 0}, {1, 2}]]
(*6.28319*)
The volume of an Ellipsoid would be
Integrate[1, {x, y, z} \[Element] Ellipsoid[{0, 0, 0}, {r1, r2, r3}], Assumptions -> {r1 > 0, r2 > 0, r3 > 0}]
(*4/3 \[Pi] r1 r2 r3*)
or e.g.
NIntegrate[1, {x, y, z} \[Element] Ellipsoid[{0, 0, 0}, {1, 2, 3}]]
(*25.1327*)