a) \(\displaystyle y = 2 - {x^2},y = 1\), quanh trục \(\displaystyle Ox\).
Ta có: \(\displaystyle 2 - {x^2} = 1 \Leftrightarrow {x^2} = 1\)\(\displaystyle \Leftrightarrow x = \pm 1\)
Khi đó \(\displaystyle V = \pi \int\limits_{ - 1}^1 {\left| {{{\left( {2 - {x^2}} \right)}^2} - 1} \right|dx} \) \(\displaystyle = \pi \int\limits_{ - 1}^1 {\left| {{x^4} - 4{x^2} + 3} \right|dx} \) \(\displaystyle = \pi \left| {\int\limits_{ - 1}^1 {\left( {{x^4} - 4{x^2} + 3} \right)dx} } \right|\)
\(\displaystyle = \pi \left| {\left. {\left( {\frac{{{x^5}}}{5} - \frac{4}{3}{x^3} + 3x} \right)} \right|_{ - 1}^1} \right|\) \(\displaystyle = \pi \left| {\frac{1}{5} - \frac{4}{3} + 3 + \frac{1}{5} - \frac{4}{3} + 3} \right| = \frac{{56\pi }}{{15}}\)
b) \(\displaystyle y = 2x - {x^2},y = x\), quanh trục \(\displaystyle Ox\).
Ta có: \(\displaystyle 2x - {x^2} = x \Leftrightarrow {x^2} - x = 0\) \(\displaystyle \Leftrightarrow x\left( {x - 1} \right) = 0 \Leftrightarrow \left[ \begin{array}{l}x = 0\\x = 1\end{array} \right.\)
Khi đó \(\displaystyle V = \pi \int\limits_0^1 {\left| {{{\left( {2x - {x^2}} \right)}^2} - {x^2}} \right|dx} \) \(\displaystyle = \pi \int\limits_0^1 {\left| {4{x^2} - 4{x^3} + {x^4} - {x^2}} \right|dx} \)
\(\displaystyle = \pi \left| {\int\limits_0^1 {\left( {{x^4} - 4{x^3} + 3{x^2}} \right)dx} } \right|\) \(\displaystyle = \pi \left| {\left. {\left( {\frac{{{x^5}}}{5} - {x^4} + {x^3}} \right)} \right|_0^1} \right|\) \(\displaystyle = \pi \left| {\frac{1}{5} - 1 + 1} \right| = \frac{\pi }{5}\)
c) \(\displaystyle y = {(2x + 1)^{\frac{1}{3}}},x = 0,y = 3\), quanh trục \(\displaystyle Oy\).
Ta có: \(\displaystyle y = {(2x + 1)^{\frac{1}{3}}} \Leftrightarrow x = \frac{{{y^3} - 1}}{2}\) với \(\displaystyle y > 0\).
Khi đó \(\displaystyle \frac{{{y^3} - 1}}{2} = 0 \Leftrightarrow {y^3} = 1 \Leftrightarrow y = 1\)
\(\displaystyle \Rightarrow V = \pi \int\limits_1^3 {{{\left( {\frac{{{y^3} - 1}}{2}} \right)}^2}dy} \) \(\displaystyle = \pi \int\limits_1^3 {\frac{{{y^6} - 2{y^3} + 1}}{4}dy} \) \(\displaystyle = \frac{\pi }{4}\int\limits_1^3 {\left( {{y^6} - 2{y^3} + 1} \right)dy} \)
\(\displaystyle = \frac{\pi }{4}.\left( {\frac{{{y^7}}}{7} - \frac{1}{2}{y^4} + y} \right)_1^3\) \(\displaystyle = \frac{\pi }{4}\left| {\frac{{{3^7}}}{7} - \frac{{{3^4}}}{2} + 3 - \frac{1}{7} + \frac{1}{2} - 1} \right|\) \(\displaystyle = \frac{{480\pi }}{7}\)
