Volumes of Revolution Help! (1 Viewer)

[alussovsky]

We recently began integration and I'm still somewhat confused about this topic of revolution volumes. For some of these questions, I can't seem to get them at all! Like I keep getting for question (a) instead of (It should be , right...?) I can't get (d) either, and keep ending up with

 

[HeroWise]

Well You are right for the first one:

Perhaps a textbook errata

and for the last one:

Sorry for the matrices didnt realise it was ther ahhaha,

 

[alussovsky]

Phew, I was so confused for the first one, haha. ^^"
AND OH GEEZ, I JUST REALISED WHAT WAS WRONG WITH THE SECOND ONE WHEN I DID IT; I FORGOT TO SQUARE THE . I'M ABSOLUTELY MORTIFIED, SORRY ABOUT THAT-
AND THANK YOU SO MUCH!

 

[HeroWise]

No problems anytime

 

[jathu123]

For part a) the region bounded by all those three 3 curves is actually the upper triangle (ie limits of integration is from 1 to 2 ). You can figure this out if you draw all those 3 curves, you'll see that only the upper triangle is fully bounded/enclosed

So subbing the new limits in we should get 1/3pi (V = 1/3pir^2h = 1/3pi)

 

[HeroWise]

I thought of that case, but its is trippy as hell then. i drew it up there, but what do you mean by fully enclosed? The bottom part is enclosed too isnt it? A little elaboration would be appreciated and thanks Jathu

 

[jathu123]

I thought of that case, but its is trippy as hell then. i drew it up there, but what do you mean by fully enclosed? The bottom part is enclosed too isnt it? A little elaboration would be appreciated and thanks Jathu

yeh it can be pretty confusing and dodgy in the beginning, but basically once you draw up all the curves, try to identify which closed region they 'form.' So once we draw all the curves/lines (in red), they'll only form a triangle on top (try not to think of the axes if it helps)

https://imgur.com/a/aspvRsw

If we were meant to find the volume generated by the bottom part, then we would also get the curve 'y = 0' along with the other 3, this will ensure that the bottom trapezium (limits from 0 to 1) is enclosed by all the curves (instead of the triangle)