Find the volume of the solid formed by the revolution of the cycloid. x=a(t-sint),y=a(1-cost), about its base

Question

Based on higher enginearing in Mathematics.

Answer 1

If the cycloid is moving upward, then it forms an inverted cone where r=x and h=y
Volume of the cone=1/3 pi x r^2 x h
If it is an Archemedian spiral, then if r is the line joining the origin and tangent to to the spiral and makes an angle theta with the origin, then the spiral rotates asymptoically and the asymptot is a = r x theta away from origin. If this is taken as the radius of the cone, then sbustituting in the above equation,
volume = 1/3 pi x r^2 x theta ^2.x a(1-cost)