Advanced Higher Maths - Volume of revolution

Hello,

I've just completed this textbook question, obtaining an answer of 6π units^3. However, the textbook answer is 2π units^3.

Would somebody be able to explain the solution, please?

(edited 1 year ago)

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Reply 1

tonyiptony

It would be helpful if you show how you get to your incorrect answer of 6pi...

Reply 2

anonymous750

Original post by tonyiptony

It would be helpful if you show how you get to your incorrect answer of 6pi...

Hopefully you can access the photo - file name 'A.jpg' in my original post

(edited 1 year ago)

Reply 3

tonyiptony

Ah, okay. V2 looks right, the problem is V1.

You've found the volume of revolution of the red region instead.

So if you want to find the volume you actually want...

EDIT: Hmm... maybe there is some error in V2 as well. I should stop doing maths while being mildly delirious, huh...

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Reply 4

mqb2766

By a simple scaling, the v1 must be v2/2, so cant help but think there is an error in the answer. A volume of 2pi just seems too small.

Reply 5

anonymous750

Yeah, it's strange. If V2 is right, then V1 would have to be negative to make 2pi. The textbook answers have contained a few errors in other chapters, as it happens.

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Reply 6

mqb2766

Original post by anonymous750

Yeah, it's strange. If V2 is right, then V1 would have to be negative to make 2pi. The textbook has had a few errors already, as it happens.

6pi is more sensible and
https://www.wolframalpha.com/input?i=volume+of+rotation+about+y+axis+sqrt%284-y%2F2%29+from+y%3D4+to+y%3D8

Reply 7

anonymous750

Original post by mqb2766

6pi is more sensible and
https://www.wolframalpha.com/input?i=volume+of+rotation+about+y+axis+sqrt%284-y%2F2%29+from+y%3D4+to+y%3D8

Thanks for confirming - maybe the textbook answer's just wrong!

Reply 8

mqb2766

Original post by anonymous750

Thanks for confirming - maybe the textbook answer's just wrong!

Yes. A volume of 6 pi corresponds to a cylinder of radius 1 and height 6 which is pretty much your shape. Its certainly not cylinder of radius 1, height 2. Cant help think they subtracted the two volumes and didnt check.

The volume of revvolution of a parabola is an old result
https://math-physics-problems.fandom.com/wiki/Volume_of_the_Paraboloid
its half the enclosing cylinder, hence the r=sqrt(2) limit on the base and heght 6. The cylinder volume is 12 pi, so the rotated parabola is 6 pi.