Volume de révolution

La cubature d'un solide est le calcul de son volume. On se limite aux solides de révolution d'axe $d$, à savoir des solides engendrés par la rotation d'une surface plane autour de la droite $d$.

L'idée est la même que lorsque l'on cherchait l'aire délimitée par le graphe d'une fonction. On subdivise l'intervalle [a,b] en n intervalles de même longueur $\frac{b-a}{n}$ et on pose $x_i=a+i\cdot\frac{b-a}{n}$ avec $x_0=a$ et $x_n=b$.

Pour chaque $0\leq i\leq n-1$, on considère le rectangle ayant comme base le segment $[x_i ; x_{i+1}]$ et comme hauteur $f(x_i)$. Chacun de ces rectangles, lorsqu'il tourne autour de l'axe $\mathrm{O} x,$ engendre un cylindre très fin de volume $\mathrm{V}_{i}=\pi\left(f\left(x_{i}\right)\right)^{2} \Delta x_{i}$ où $\Delta x_{i}=x_{i+1}-x_{i}$.

Finalement, par passage à la limite, on obtient : $$\mathrm{V}=\lim _{n \rightarrow+\infty} \sum_{i=0}^{n-1} \mathrm{~V}_{i}=\lim _{n \rightarrow+\infty} \sum_{i=0}^{n-1} \pi\left(f\left(x_{i}\right)\right)^{2} d x_{i}=\pi \lim _{n \rightarrow+\infty} \sum_{i=0}^{n-1}\left(f\left(x_{i}\right)\right)^{2} d x_{i}$$ Dès lors, si l'on considère une rotation autour de l'axe $x$ d'une surface plane délimitée par le graphe de l'équation $y = f(x)$, avec $f$ intégrable sur $[a, b]$, et les droites d'équations $x = a$ et $x = b$, le volume du solide de révolution se calcule par : \[V = \pi \int_a^b f^2(x) \ \textrm{d}x\]

Soient deux fonctions $f\left( x \right)$ et $g\left( x \right)$ continues, positives sur un intervalle $\left[ {a,b} \right]$ avec $g\left( x \right) \le f\left( x \right)$. La formule générale pour calculer le volume V d'un solide de révolution engendré par la rotation d'une région située entre deux courbes $f(x)$ et $g(x)$ entre $x=a$ et $x=b$ autour d'un axe des abscisses est donnée par : \[\boxed{V=\pi \int_{a}^{b} f^2(x)-g^2(x) \ \textrm{d}x}\]

Source Tikz de la figure

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\end{document}

Exemple

\begin{align*} V &= \pi \int_1^4 {\left( {{\left( {\sqrt{x}} \right)^2} - {1^2}} \right)dx} \\ &= \pi \int_1^4 {\left( x - 1 \right)dx} \\ &= \pi \left. {\left( {\frac{{x^2}}{2} - x} \right)} \right|_1^4 \\ &= \pi \left( {\left( {\frac{{4^2}}{2} - 4} \right) - \left( {\frac{{1^2}}{2} - 1} \right)} \right) \\ &= \frac{9\pi}{2} \end{align*} source Figure 6.2.12

Exemple

Le volume du solide de révolution engendré par la rotation de la région du plan délimitée par les équations $y=x^2$ et $y=\sqrt{x}$ est donnée par : \[\begin{aligned}[t] V = \pi \int_0^1 {\left( {{{\left[ {\sqrt x } \right]}^2} - {{\left[ {{x^2}} \right]}^2}} \right)dx} &= \pi \int_0^1 {\left( {x - {x^4}} \right)dx}\\ &= \pi \left. {\left( {\frac{{{x^2}}}{2} - \frac{{{x^5}}}{5}} \right)} \right|_0^1\\ &= \pi \left( {\frac{1}{2} - \frac{1}{5}} \right) = \frac{{3\pi }}{{10}} \end{aligned}\]

Volume de révolution

Solide de révolution engendré par la rotation d'une région située entre deux courbes