Solid of Revolution 3D Visualizer – AP Calculus

Learning Objectives

Visualize the 3D volume generated by revolving a 2D area

Compare Disk and Washer integration methods

Connect Riemann sums ($\Delta x$) to cylindrical slices

Observe convergence as $n \to \infty$ or $\Delta x \to 0$

Key Equations

$ \text{Disk: } V = \pi \int_{a}^{b} [R(x)]^2 \, dx $

$ \text{Washer: } V = \pi \int_{a}^{b} ([R(x)]^2 - [r(x)]^2) \, dx $

$ \text{Shell: } V = 2\pi \int_{a}^{b} p(x) h(x) \, dx $

Why It Matters

Integrating abstract geometric shapes can be unintuitive. By generating the transparent 3D slices directly from the 2D function curve, you can literally see the discrete "disks" summing up into a smooth, continuous solid object.