I'm reading the book Mathematics: It's Content, Methods, and Meanings and I'm unsure as to how one of the variables in an example was derived. The question is about the volume of a paraboloid and here is the text and diagram. image How is the the radius the sqr (z)? It claims it is quite obvious but I'm unable to understand. I think the volume of the bottom cylinder is the area of the bottom circle times delta z and the volume of the top cylinder is the area of top circle times delta z but that is just a guess and still leaves me the question of how the radius was derived from the height.

Thanks, Jackson

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    $\begingroup$ The usual formula for a paraboloid is $z=x^2+y^2$. Convince yourself the graph of this equation is the same as the one in the picture; then, in cylindrical coordinates, you have $z=r^2$. $\endgroup$ – Clayton May 10 '15 at 1:23

You could better understand the argumentation if you imagined the $z$ axis horizontally and would not look at the whole solid body but only its cross section with the $zy$ plane. The you would see the following image:

enter image description here

$\Delta V$ is one slice, a disk, of the paraboloid.


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