| Version 8 |
Version 7 |
| \emph{Torricelli's trumpet} is a fictional infinitely long solid of revolution formed when the closed domain |
\emph{Torricelli's trumpet} is a fictional infinitely long solid of revolution formed when the closed domain |
| $$A := \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; x \ge 1,\; 0 \le y \le \frac{1}{x}\}$$ |
$$A := \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; x \ge 1,\; 0 \le y \le \frac{1}{x}\}$$ |
| rotates about the $x$-axis. It has a finite volume, $\pi$ volume \PMlinkescapetext{units}, but the area of its surface is infinite; in fact even the area of $A$ is infinite, i.e., the improper integral $\displaystyle\int_1^\infty\frac{1}{x}\,dx$ is not convergent. |
rotates about the $x$-axis. It has a finite volume, $\pi$ volume \PMlinkescapetext{units}, but the area of its surface is infinite; in fact even the area of $A$ is infinite, i.e., the improper integral $\displaystyle\int_1^\infty\frac{1}{x}\,dx$ is not convergent. |
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Torricelli's trumpet is surprising since the the solid can be filled by a finite amount
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Torricelli's trumpet is paradoxical since the the solid can be filled by a finite amount
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| of paint, but this paint can never suffice for painting the surface of the body, no matter |
of paint, but this paint can never suffice for painting the surface of the body, no matter |
| how \PMlinkescapetext{thin} a coat of paint is used! |
how \PMlinkescapetext{thin} a coat of paint is used! |
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| \PMlinkescapeword{even} |
\PMlinkescapeword{even} |
