gen-define-volume-of-solids-of-revolution-and-give-one-short-wor-3bae

Define volume of solids of revolution and give one short worked example.

# Volume of Solids of Revolution ## Definition A **solid of revolution** is a 3D object formed by rotating a 2D region about an axis (usually the x-axis or y-axis). The volume can be calculated using: **Disk Method:** When rotating about an axis: $$V = \pi \int_a^b [R(x)]^2 \, dx$$ where R(x) is the radius function (distance from the axis to the curve). --- ## Worked Example **Problem:** Find the volume when the region bounded by $y = x^2$, $y = 0$, and $x = 2$ is rotated about the x-axis. **Solution:** 1) **Identify the radius:** $R(x) = x^2$ (distance from x-axis to curve) 2) **Set up the integral:** $$V = \pi \int_0^2 (x^2)^2 \, dx = \pi \int_0^2 x^4 \, dx$$ 3) **Evaluate:** $$V = \pi \left[\frac{x^5}{5}\right]_0^2 = \pi \left(\frac{32}{5} - 0\right) = \frac{32\pi}{5}$$ **Answer:** $V = \frac{32\pi}{5}$ cubic units (≈ 20.11 cubic units)