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I'm trying to find the volume of a solid that consists of a square-base pyramid and a paraboloid.

Solid in question

I am given $z$ in terms of $r$ for both of solids.

My first thought is dividing the solid into two separate ones and then using triple integrals to find their individual volumes, so I can add them. However, I am not used to having $z$ defined this way, so I am quite confused in how to approach this problem. Any help would be appreciated.

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1 Answer 1

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Looking at your figure I see neither a pyramid nor a paraboloid. Instead, it shows a picture of the set $$B:=\bigl\{(x,y,z)\bigm| -\sqrt{1-x^2-y^2}\leq z\leq 1-\sqrt{x^2+y^2}\bigr\}\ ,$$ whereby the letter $r$ has been interpreted as $r:=\sqrt{x^2+y^2}$. The set $B$ is the union of a half ball and a cone of height $1$ glued across the unit disc, hence $${\rm vol}(B)={2\pi\over3}+{\pi\over3}=\pi\ .$$ Archimedes could have done this.

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