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Revision difference : Torricelli's trumpet |
| Version 3 |
Version 2 |
| The {\em Torricelli's trumpet} is fictional infinitely long body of revolution formed when the closed domain |
The {\em Torricelli's trumpet} is fictional infinitely long body 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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| The paradoxality of the Torricelli's trumpet may be illustrated so that the body may be filled by a finite amount of paint, but this paint does never suffice for painting the surface of the body, insignificant how \PMlinkescapetext{thin} is the paint coat! |
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