# Volumes
###### tags: `Calculus`
### Cavalieri's Priciple
$$
If\;a\;family\;of\;parallel\;planes\;gives\;equal\;cross-sectional\;areas\;\\for\;two\;solids,then\;they\;have\;the\;same\;volume
$$
##
### Volume rotate on y=-k
$$
V_0=\int_a^b2\pi y\cdot l(y)dy\\V_k=\int_a^b2\pi(y+k)\cdot l(y)dy\\=\int_a^b2\pi y\cdot l(y)dy+\int_a^b2\pi kl(y)dy\\=V_0+2\pi k\int_a^bl(y)dy\\=V_0+2\pi kA
$$