Question

Show the formula of surface when curve revolve about theta =0 line

Answer 1

I have the cardioid r=1+cos(t)r=1+cos(t) for 0≤t≤2π0≤t≤2π and I want to calculate the surface of revolution of said curve. How can I calculate it?

The parematrization of the cardioid is:

x(t)=(1+cos(t))cos(t)x(t)=(1+cos(t))cos(t)

y(t)=(1+cos(t))sin(t)y(t)=(1+cos(t))sin(t)

and

dxdt=(−2cos(t)−1)sin(t)dxdt=(−2cos⁡(t)−1)sin⁡(t)

dydt=cos(t)(cos(t)+1)−sin2(t)dydt=cos⁡(t)(cos⁡(t)+1)−sin2⁡(t)

To calculate the surface of revolution I know I can use the formula (since I want to revolve it around the x-axis)

2π∫bay(t)(dxdt)2+(dydt)2−−−−−−−−−−−−−−√dt