calculate by double integration the volume generated by the revolution of the cardioid r = a (1 – cos ď±) about its axis
Answers
The parametric equations of a cardioid are x=cosθ(1−cosθ) and y=sinθ(1−cosθ), 0≤θ≤2π. Diagram here. The region enclosed by the cardioid is rotated about the x-axis, find the volume of the solid. I am not allowed to use polar form, or double integrals, due to the limitations of the NSW mathematics syllabus. Basically I'm stuck with using a disk approximation. What I've got so far using the disk approximation is:
limδx→0∑0x=−2πy2δx=π∫0−2y2dx=π∫π2πsin2θ(1−cosθ)2(2cosθsinθ−sinθ)dθ which gives the volume of the portion from x= -2 to 0, but I have no idea how to calculate the volume from x = 0 to 1/4. My gut feeling says to just extend the bounds of the previous integral to π and 0, but because there exists two y-values for each x value in that domain, shouldn't I have to subtract the volume of the larger disk from the smaller disk i.e. form an annulus?