About integration taking the sphere volume as an example (find the error)

Question

I have heard a lot about rotation and how you can use it to calculate the volumes of different bodies.
There is something though which has troubled me for a long time.
If you take the area of half a disk : $\mathscr{A}/2 = \frac{\pi r^{2}}{2}$ and that you argue that by integrating it from $0$ to $2\pi$ you will have accomplished a full rotation and should (as done with the volume of cones) now get the whole sphere, you arrive to the following result: $$\int_{0}^{2\pi} \mathscr{A}/2 \; d\theta = \pi^{2}r^{2} $$ Which is obviously false.
Now here is my question, where did I do something wrong ?