Biraketa-solido

${\displaystyle y=f(x)}$, non ${\displaystyle x\in \langle a,b\rangle }$, kurbaren biraketa-solidoaren bolumena X ardatzaren inguruan:

f(x)

${\displaystyle V=\pi \int \limits _{a}^{b}(f(x))^{2}\,dx}$

Azalera:

${\displaystyle S=2\pi \int \limits _{a}^{b}|f(x)|{\sqrt {1+[f'(x)]^{2}}}\,dx}$

Funtzioa adierazpide parametrikoan denean

${\displaystyle x=x(t),\ y=y(t);\ t\in \langle t_{1},t_{2}\rangle }$

Bolumena:

${\displaystyle V=\pi \left|\int \limits _{t_{1}}^{t_{2}}y^{2}(t)x'(t)\,dt\right|}$

Azalera:

${\displaystyle S=2\pi \int \limits _{t_{1}}^{t_{2}}|y(t)|{\sqrt {[x'(t)]^{2}+[y'(t)]^{2}}}\,dt}$

• Biraketa-gainazala
• Pappus-Guldinen teorema