The curve C has parametric equations x=cos(t)+1/2*sin(2t) and y =-(1+sin(t)) for 0<=t<=2π. Find a Cartesian equation for C. Find the volume of the solid of revolution of C about the y-axis.

Note the simplest relation to eliminate t is the fact cos²(t)+sin²(t)=1 for all t, so we need only find x and y in terms of cos(t) and sin(t).Note we have sin(t)=-(y+1) from the equation for y already.From the equation for x, x=cos(t)+1/2sin(2t). The key step is to use the double angle formulae to express sin(2t) in terms of sin(t) and cos(t). sin(2t)=2sin(t)*cos(t) gives x=cos(t)+sin(t)cos(t) = cos(t)(1+sin(t)). We recognise 1+sin(t)=-y and so x=-ycos(t) gives cos(t)=-(x/y).Then cos²(t)+sin²(t)=1 for all t => (-x/y)^2+(-y-1)^2=1 => x^2/y^2 = -y^2-2y => x^2=-(y^4+2y^3).
The volume of the pendant is calculated from the formulae π∫x^2 dy from y = 0 -> y = -2 (from sketch), calculated with above expression for x^2.