y=sinx,x=π/2,y=0,绕y轴,求旋转体的体积把公式带进去,

Question

y=sinx,x=π/2,y=0,绕y轴,求旋转体的体积
把公式带进去,

Answer 1

x = 0到x = π/2?
V = 2π∫(0→π/2) xsinx dx,by shell method
= 2π∫(0→π/2) x d(- cosx)
= - 2πxcosx |(0→π/2) + 2π∫(0→π/2) cosx dx
= 2πsinx |(0→π/2)
= 2π
π∫(0→1) (arcsiny)² dy,by disc method
= πy(arcsiny)² |(0→1) - π∫(0→1) y • 2arcsiny • 1/√(1 - y²) dy
= π(π/2)² - 2π∫(0→1) yarcsiny/√(1 - y²) dy,y = sint,dy = cost dt
= π³/4 - 2π∫(0→π/2) tsint/cost • cost dt
= π³/4 - 2π∫(0→π/2) t d(- cost)
= π³/4 + 2πtcost |(0→π/2) - 2π∫(0→π/2) cost dt
= π³/4 - 2πsint |(0→π/2)
= π³/4 - 2π
∴V = π(π/2)²(1) - (π³/4 - 2π) = 2π