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The question is in the following picture:

enter image description here

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!

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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.

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  • $\begingroup$ Why he approximated the base area by 3? this point is not clear for me. $\endgroup$ – user426277 Sep 8 '17 at 8:43
  • $\begingroup$ why the upper surface of the solid is not horizontal ? is this because of the "y" in y+3? is the upper surface of the cylinder oblique? $\endgroup$ – user426277 Sep 8 '17 at 8:47
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    $\begingroup$ The area estimate is clearly explained in the video: he argues that the space between those two parabolas takes more than half of a $2\times2$ square, hence its area should be something more than $2$, probably around $3$. And yes, the upper face of the cylinder is oblique: it would have been horizontal if its equation had been $z=3$. $\endgroup$ – Aretino Sep 8 '17 at 9:18

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