Here they are asking me to use divergence theorem to calculate this integral. I know that to be able to use divergence theorem, we need a closed surface so that it has a volume. Thus in my understanding, we should add two flat disks in the top and in the bottom of the cylidner so that it becomes closed so that we can use divergence theorem. Then, after we compute the triple integral, we SUBSTRACT the surface area of the top disk and the bottom disk, so that we have the good answer. However, in the answer key, the teacher just assigned normal vectors pointing outwards of the two disk ( instead of inwards for some reason ) and she decided to add the two surface integrals to the triple integral instead of substracting them. I think there's something I dont understand since I tought that you need to subract the surface areas since they are extra. why is she adding them ? Down below is the question itself, and a part of the answer key.
You're fine. The inward-pointing normal on the cylinder $C$ accounts for the negative sign. The answer key did not orient the two disks compatibly with the cylinder, but kept them with an outward-pointing normal. If you proceed with your approach and subtract the flux across the two disks — oriented inward — then you will have the identical result. (Subtracting the flux with the inward orientation is adding the flux with the outward orientation.)
However, remember that that you need the outward orientation on the surface in order to apply the Divergence Theorem. So ... the approach of solution key is the best way to do the problem.