Works in my copy of 11.3

In[6]:= Assuming[a > 0 && b > 0, Integrate[1, {x, y} \[Element] Circle[{0, 0}, {a, b}]]]

Out[6]= 2 (b EllipticE[1 - a^2/b^2] + a EllipticE[1 - b^2/a^2])

Yes with Assumptions work in my copy to.

Thanks.

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*)