Question

calcular o volume do solido abaixo utilizando o método do anel.

Answer 1

$y=3x-x^{2}$

$V= \pi \int\limits^a_b {f(x)^{2}} \, dx \\V= \pi \int\limits^3_0 {(3x-x^{2})^{2}} \, dx\\V=\pi \int\limits^3_0 {9x^{2}-6x^{3}+x^{4}} \, dx\\V= \pi[ [ \frac{9x^{3}}{3} ]^{3} _{0}-[ \frac{6x^{4}}{4} ]^{3}_{0}+[ \frac{x^{5}}{5} ]^{3}_{0} ]\\V= \pi[ (\frac{9*3^{3}}{3} )-( \frac{6*3^{4}}{4} )+( \frac{3^{5}}{5}) ]\\V= \pi[81-121,5+48,6]\\V=8,1 \pi (u.v)$