Overview
This lecture explains how to find the volume of solids using integration, specifically focusing on the disc and washer methods for solids of revolution, with detailed steps, formulas, and example problems.
Volume by Slicing & Cross-Section Method
- Volume can be found by slicing a solid into thin slabs and adding their volumes together.
- For rectangles, volume = base area × length; for general shapes, volume = cross-sectional area × thickness.
- Approximating the solid as many thin slabs leads to a Riemann sum and, in the limit, an integral.
- The general formula: V = ∫[A to B] (cross-sectional area) dx for solids with cross-sections perpendicular to the x-axis.
Solids of Revolution: Disc Method
- Revolving a function around an axis creates a solid with circular cross-sections (discs).
- For rotation about the x-axis, each cross-section is a circle with radius equal to the function’s value at x.
- Formula: V = ∫[A to B] π[f(x)]² dx
- The radius r of each disc = f(x); πr² is the cross-sectional area.
Example: Find the volume of the solid formed by revolving y = 3√x from x = 1 to x = 4 about the x-axis.
- Setup: V = ∫[1 to 4] π(3√x)² dx = ∫[1 to 4] 9πx dx
Solids of Revolution: Washer Method
- Used when revolving a region bounded by two functions, f(x) (outer) and g(x) (inner), about an axis, forming a "washer" (circle with a hole).
- Cross-section area = π[outer radius² – inner radius²].
- Formula: V = ∫[A to B] π([f(x)]² - [g(x)]²) dx
Washer Example
- For functions f(x) and g(x) on [a, b]:
- V = ∫[a to b] π(f(x)² - g(x)²) dx
Volumes of Revolution Around the y-Axis
- When rotating around the y-axis, functions must be written as x = f(y), bounds are in y.
- Disc: V = ∫[C to D] π[u(y)]² dy
- Washer: V = ∫[C to D] π([f(y)]² - [g(y)]²) dy
Volumes Revolved Around Lines Other Than Axes
- When rotating around y = c (horizontal line), the radius = |c - function(x)| for each function involved.
- Setup the washer/disc method as above, using adjusted radii.
Key Terms & Definitions
- Cross-sectional area — Area of a "slice" perpendicular to an axis within a solid.
- Disc method — Technique for finding volume by summing the volumes of infinitesimally thin discs.
- Washer method — Modification of the disc method for solids with holes; uses two functions to determine inner and outer radii.
- Solid of revolution — A solid formed by revolving a region around a line (axis).
- Riemann sum — Sum used to approximate integrals, becomes exact in the limit as the partition gets finer.
Action Items / Next Steps
- Practice setting up and evaluating disc and washer integrals for various axes and functions.
- Ensure comfort switching between x and y as the variable of integration based on the axis of rotation.
- Review homework problems on volumes by the disc and washer methods.
- Prepare for the upcoming section on the cylindrical shell method.