1、旋转体体积是V=∫[-1/2,1] π(2x+1)^2dx=π/6(2x+1)^3 [-1/2,1]=9π/2
2、体积为V=∫[-√3,√3]π(√4-x^2-1)^2dx
∫(√4-x^2-1)^2dx=∫5-x^2-2√4-x^2-1dx=5x-1/3x^3-2∫√4-x^2dx
令x=2sinu dx=2cosudu
∫√4-x^2dx=4∫cos^2udu=2∫(1+cos2u)du=2u+sin2u=2arcsin(x/2)+x√4-x^2/2
V=∫[-√3,√3]π(√4-x^2-1)^2dx=π (5x-1/3x^3-4arcsin(x/2)-x√4-x^2) [-√3,√3]=π(6√3-8π/3)