Bonjour,
A)
[tex]1)\ V= \dfrac{4 \pi R^3}{3} \\\\
2) Th\'eor\`eme\ de \ Pythagore: r^2(z)=R^2-z^2\\\\
3) V=base*hauteur= \pi (R^2-z^2)*\Delta z\\\\
4) \int\limits^{R}_{-R} {\pi (R^2-z^2)} \, dz = [\pi (R^2z- \dfrac{z^3}{3} )]_{-R}^R\\\\
=\dfrac{2 \pi R^3}{3}
[/tex]
B)
[tex]1) V= \dfrac{ \pi R^2h}{3} \\\\
2)En \ utilisant\ Thal\`es, \dfrac{r(z)}{R} = \dfrac{h-z}{h} \\\\
3) \Delta V= \pi r^2(z)*\Delta z= \pi \dfrac{(R(h-z))^2}{h^2} \Delta z\\\\
4) \int\limits^h_0 {\pi \dfrac{(R(h-z))^2}{h^2} } \, dz\\\\
=\dfrac{ \pi R^2}{h^2} *[- \dfrac{(h-z)^3}{3} ]_0^h\\\\
= \dfrac{ \pi R^2[0+h^3)}{3*h^2} \\\\
=\dfrac{\pi R^2*h}{3}
[/tex]
