The question is in the following picture:
The solution of the question is in the following video, https://www.youtube.com/watch?v=2zry1DM7I8U&list=PL81IATpFpPBgrG8fZ3tRO41nNypY5xtEP&index=22, by integration, but the lecturer in the video mentioned the solution by areas, he guesses that the area is 9 but I did not understand why he guesses this. Could anyone explain for me this method please? Thanks!
He approximates the solid with a (generalized) cylinder, having a base of area around $3$ and a height of $3$. I wrote "generalized" cylinder, because you are probably only familiar with cylinders whose base is a circle, but the actual shape of the base is not important: the volume of the cylinder can be always computed as base area $\times$ height.
This estimate is actually a lower bound, because the upper surface of the solid is not horizontal, and its distance from the cylinder base is $\ge3$. So the lecturer can guess that the volume of the cylinder must be somewhat greater than $9$: that is enough to choose the right answer among the given alternatives.