Bài tập 3.59 trang 184 SBT Toán 12

Ta có: \(\displaystyle  V = \pi \int\limits_0^{\frac{\pi }{2}} {{{\left( {{{\sin }^{\frac{3}{2}}}x} \right)}^2}dx} \) \(\displaystyle   = \pi .\int\limits_0^{\frac{\pi }{2}} {{{\sin }^3}xdx} \) \(\displaystyle   = \pi .\int\limits_0^{\frac{\pi }{2}} {\left( {1 - {{\cos }^2}x} \right)\sin xdx} \)

\(\displaystyle   =  - \pi .\int\limits_0^{\frac{\pi }{2}} {\left( {1 - {{\cos }^2}x} \right)d\left( {\cos x} \right)} \) \(\displaystyle   =  - \pi .\left. {\left( {\cos x - \frac{{{{\cos }^3}x}}{3}} \right)} \right|_0^{\frac{\pi }{2}}\) \(\displaystyle   =  - \pi \left( { - 1 + \frac{1}{3}} \right) = \frac{{2\pi }}{3}\)

-- Mod Toán 12 HỌC247