∫y√(2y-y²)dy
=∫y√(2y-y²-1+1)dy
=∫y√(1-(y-1)²)dy
设y-1=sintcost=√(1-(y-1)²)sin2t=2sintcost=2(y-1)√(1-(y-1)²)
y=sint+1dy=costdt
原式=∫(sint+1)costcostdt
=∫sintcos²tdt+∫cos²tdt
=-∫cos²tdcost+∫(1+cos2t)/2dt
=-(cos³t)/3+t/2+(sin2t)/4+C
=-(1/3)*√(1-(y-1)²)³+(1/2)*arcsin(y-1)+(1/2)*(y-1)√(1-(y-1)²)+C