Overview
This lecture explains how to find the volume of a solid formed when an area under a curve is rotated 360° about the x-axis, using integration.
Solids and Volumes of Revolution
- A solid of revolution is formed by rotating a region bounded by a function and boundary lines around the x-axis.
- The resulting 3D shape's volume is called the volume of revolution.
Finding the Volume Around the x-Axis
- To calculate the volume when the area under y = f(x), from x = a to x = b, is rotated about the x-axis, use the formula:
- ( V = \pi \int_{a}^{b} y^2 dx )
- y is a function of x, so substitute accordingly before integrating.
- π is a constant; it can be placed inside or outside the integral.
- This formula is not given in the standard formulae booklet.
Steps to Solve Volume of Revolution Problems
- Visualize or sketch the function and the resulting solid to understand the problem.
- Step 1: Square the expression for y before integrating.
- Step 2: Identify the limits a and b, either from equations or a graph.
- Step 3: Evaluate the definite integral and multiply by π.
- Give the answer in exact form (with π) unless asked to round, usually to three significant figures.
- Sometimes, you'll be given the volume and asked to find unknowns in the function.
Examiner Tips and Memory Aids
- The formula involves ( y^2 ) because each slice perpendicular to the x-axis is a circle with area ( \pi y^2 ).
- Integrating adds up all circular "slices" from a to b, creating the full volume.
Key Terms & Definitions
- Solid of Revolution — 3D shape formed by rotating a bounded area about an axis.
- Volume of Revolution — The volume of a solid formed by such rotation, found by integration.
- Limits (a, b) — The x-values defining the interval of integration.
- Definite Integral — An integral with upper and lower limits, used to calculate area or volume.
Action Items / Next Steps
- Practice using ( V = \pi \int_{a}^{b} [f(x)]^2 dx ) with sample functions and limits.
- Sketch functions and their solids of revolution for better understanding.
- Remember to leave answers in terms of π, unless otherwise specified.