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It is quite possible the tooth is still strong and without fracture. It is possible that the swelling & pain (probable infection) is due to an issue other than fracture. If so, the tooth may be able to be saved - possibly without additional(or minimal) treatment. But if it is fractured, it likely will need to be removed. It's a matter of cost - benefit for you - whether or not you feel the 3D CT is worth the cost to determine if the tooth can be saved, or whether you'd rather not pursue diagnosing a possibly non-restorable tooth. It's a judgement call. I thin that an xray - possibly a 3D CT is worthwhile to determine whether or not there is a fracture before extracting the tooth, depending on the expense. But the decision should certainly be yours.
Please let me know if you need additional information to answer your question I'm here to help - and happy to do so.