∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 3 Theory

Volumes of Revolution (Disks, Washers & Shells in 3D)

Revolving 2D planar curves around coordinate axes to generate 3D solids

Revolve y = f(x) through a sweep angle θ to see circular cross-sections synthesize a solid of revolution in full 3D WebGL.

1. Disk vs. Cylindrical Shell Integration

Disk integration stacks perpendicular circular slices of volume π R² dx. Shell integration unrolls nested concentric cylinders of volume 2π (radius)(height) dx.

dVdisk=π[R(x)]2 dx,dVshell=2πx f(x) dxdV_{\text{disk}} = \pi [R(x)]^2\,dx, \qquad dV_{\text{shell}} = 2\pi x\,f(x)\,dx

Quick Reference & Cheat Sheet

Solid of Revolution Formulas
Vx=π∫ab[R(x)2−r(x)2] dx,Vy=2π∫abx(f(x)−g(x)) dxV_x = \pi \int_a^b [R(x)^2 - r(x)^2]\,dx, \quad V_y = 2\pi \int_a^b x (f(x) - g(x))\,dx
Sums planar discs or cylindrical blankets around the axis.