Guldin or it's not necessary? "Surface" of revolution 2

Calculus Level pending

Let the square given by $\{(x,y) \in \mathbb{R}^2 ;1 \leq x \leq 3,\quad 1 \leq y \leq 3\}$ rotating around $x$ -axis in $\mathbb{R}^3$ .

The volume of the "surface" of revolution obtained can be written as $A\pi$ .

Submit $A$ .

The answer is 16.

1 solution

Andreas Wendler
Apr 4, 2016

Guldin tells us that volume of rotation's body is equal product of generating surface (square) and way of centroid which has y-coordinate 2:

$V= 2 \times 2 \times 2\pi \times 2 = 16\pi$

Yes,that is, thanks (+1) $\uparrow$ . This problem can also be solved as a difference of the volume of two cylinders....

Guillermo Templado - 5 years, 2 months ago

Of course! But not so ellegant. ;-)

Andreas Wendler - 5 years, 2 months ago

Ellegance is relative... I think, for most people is simplier the 2º method.

Guillermo Templado - 5 years, 2 months ago

@Guillermo Templado – So, to answer your question, (Habakkuk) Guldin is sufficient but not necessary ;) Personally, I always like to see the good work of a countryman being used

Otto Bretscher - 5 years, 2 months ago

@Otto Bretscher – Otto, I forgot to mention that there is a very entertaining book (It is a novel and it's not absolutely historic about Archimedes) It's The Sand Reckoner . I know my link is in spanish, but it's very entertaining. It might interest you, I don`t know...

Guillermo Templado - 5 years, 2 months ago

@Guillermo Templado – Thank you so much, Compañero! The book gets great reviews, and I will surely get it (in English, though). My love for the history of math as well as algebra was kindled by one of my great teachers at the University of Zürich, Baertel van der Waerden.

We love Spain and visit often (I celebrated a special birthday earlier this year at Alhambra), but my Spanish is pretty weak, although I can often figure it out with my background in Latin and my conversational French.

Otto Bretscher - 5 years, 2 months ago

@Otto Bretscher – It is amazing to me that in a time where the arithmetic was so poorly developed (zero was not clear or limited numbers... ), have such great abilities for geometry or engineering... Recuerdos de la Alhambra, (Memories of Alhambra) . Van der Waerden? he sounds me(the football soccer player,haha)... Noether?, I don't know right now..

Guillermo Templado - 5 years, 2 months ago

Guldin or it's not necessary? "Surface" of revolution 2

The answer is 16.

1 solution

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