Dunking it in water doesn't work-- you only get the volume of the rat-like creature that lives inside the cat; much like the feeble alien within a Dalek. (And, if your answer had anything to do with contour integrals, get real.)
Here is a low-tech method that works: Using successive approximation, determine the smallest box that the cat will fully enclose itself in, and measure the size of that box. Cats tend to leave a few appendages hanging out of the corners-- you may need to assist with folding the cat into the box for the final stages of approximation.
This cat is approximately 648 cubic inches in volume.