surfaces and volumes of revolution

Rotate a curve in the half-plane about the -axis.

volume (disk method)

In cylindrical coordinates . Integrating and :

Each slice at height is a disk of radius , area , thickness .

V = jnp.pi * jnp.trapezoid(jax.vmap(rho_fn)(z_grid)**2, dx=dz)

surface area

Arc length along the curve: .

Each segment traces a ribbon of circumference :

Parametric form : .

drho = jax.vmap(jax.grad(rho_fn))(z_grid)
S = 2*jnp.pi * jnp.trapezoid(rho_vals * jnp.sqrt(1 + drho**2), dx=dz)

example: cone

Line from to :