Volumes by Discs and Washers

Overview

• Topic: Volumes of solids using the methods of slicing: Discs and Washers.
• Goal: Use Riemann-sum idea to derive integrals that compute volumes by adding cross-sectional areas.
• Main methods covered: Volume by slicing (general), Discs (solid of revolution about axis), Washers (hollow solids), and application to rotations about x- and y-axes and about horizontal lines.

Key Concepts

• Volume by slicing:
  • Cut a solid into thin slabs (slices) perpendicular to an axis.
  • Volume of one slab ≈ (cross-sectional area at xk) * Δx.
  • Sum slabs and take limit as Δx → 0 to get integral: V = ∫[a→b] A(x) dx.
• Cross-sectional area:
  • For slices perpendicular to x-axis: A(x) is area of cross-section at x.
  • For slices perpendicular to y-axis: A(y) is area of cross-section at y.
• Discs method (solids of revolution without hole):
  • Cross-section is a circle with radius r(x) = f(x) when revolving y = f(x) about x-axis.
  • Formula: V = ∫[a→b] π [f(x)]^2 dx.
  • If revolving about y-axis, use functions expressed as x = u(y): V = ∫[c→d] π [u(y)]^2 dy.
• Washers method (solids of revolution with hole):
  • Outer radius R(x) from the axis, inner radius r(x) from axis.
  • Cross-sectional area = π[R(x)]^2 − π[r(x)]^2 = π( R^2 − r^2 ).
  • Formula: V = ∫[a→b] π ( [outer radius]^2 − [inner radius]^2 ) dx (or dy if perpendicular to y).
• Important setup rules:
  • Discs/Washers: express functions in terms of the variable corresponding to the axis perpendicular to slices (revolve about x-axis → functions in x; revolve about y-axis → functions in y).
  • Identify interval [a,b] by intersection points of bounding curves or given bounds.
  • Determine which curve gives the outer radius (top or right as appropriate) and which gives inner radius (bottom or left).
  • Simplify algebra before integrating.*

Formulas and Definitions

• Volume by slicing (general): V = ∫[a→b] A(x) dx.
• Disc method (about x-axis): V = ∫[a→b] π [f(x)]^2 dx.
• Disc method (about y-axis): V = ∫[c→d] π [u(y)]^2 dy.
• Washer method (about x-axis): V = ∫[a→b] π ( [F(x)]^2 − [G(x)]^2 ) dx (F outer, G inner).
• Washer method (about y-axis): V = ∫[c→d] π ( [U(y)]^2 − [V(y)]^2 ) dy.

When revolving about a horizontal line y = k:

• Outer radius = distance from line y = k to outer curve = |k − y_outer(x)|.
• Inner radius = distance from line y = k to inner curve = |k − y_inner(x)|.
• Then use washers with those radii squared and integrated over x (if slices perpendicular to x).

Worked Example Summaries

• Example 1 (simple cylinder-like stack):

  • Region: cross-section is a circle of radius 1, x from 1 to 5.
  • A(x) = π(1)^2 = π → V = ∫[1→5] π dx = 4π.
  • Matches cylinder volume formula base area * height.

• Example 2 (disc method, y = 3√x from x=1 to 4, about x-axis):

  • Setup: V = ∫[1→4] π [3√x]^2 dx = ∫[1→4] 9π x dx.
  • Integrate: 9π * (x^2/2)|1^4 = (9π/2)(16 − 1) = (9π/2)*15 = 135π/2.

• Example 3 (derive sphere volume by discs):

  • Start with upper semicircle: y = √(R^2 − x^2), x ∈ [−R, R], revolve about x-axis.
  • Disc radius = y = √(R^2 − x^2).
  • V = ∫[−R→R] π (R^2 − x^2) dx.
  • Integrate and simplify to V = (4/3)π R^3.

• Example 4 (washer method, region between f(x) and g(x) rotated about x-axis):

  • Given: region between y = f(x) and y = g(x) for x ∈ [a,b]. If f(x) ≥ g(x).
  • V = ∫[a→b] π( f(x)^2 − g(x)^2 ) dx.
  • Worked numeric: region between y = x^2 + 1/4 and y = x^2 for x ∈ [0,2] led to simplified integrand and final numeric volume (including factor π).

• Example 5 (disc about y-axis using x = u(y)):

  • y = x with y ∈ [0,2], revolve about y-axis.
  • Solve for x in terms of y: x = y.
  • V = ∫[0→2] π [x(y)]^2 dy = ∫[0→2] π y^2 dy = (π * y^3 / 3)|0^2 = (8π/3)? (Note: specific numeric result in lecture was 32π/5 for a different arrangement; always check algebra and limits for each configuration).

• Example 6 (revolving a region about a horizontal line y = 3/2):

  • Region between y = x and y = x^3 for x ∈ [0,1], revolve about y = 3/2.
  • Outer radius = 3/2 − y_inner? Determine which curve is farther from line:
    • Outer radius = |3/2 − x^3| (outer curve measured from axis).
    • Inner radius = |3/2 − x|.
  • Use washers: V = ∫[0→1] π( [3/2 − x^3]^2 − [3/2 − x]^2 ) dx.
  • Simplify algebra first before integrating; combine like terms to reduce work.

Key Problem-Solving Steps (school style)

• 1. Read the revolving axis and identify whether slices are perpendicular to x or y.
• 2. Express bounding functions in terms of the variable matching the slicing direction:
  • revolve about x-axis or horizontal line → express y = f(x) (use dx).
  • revolve about y-axis or vertical line → express x = g(y) (use dy).
• 3. Find interval of integration by solving intersections or using given bounds.
• 4. Decide method:
  • Disc (no hole): single radius → A = π [radius]^2.
  • Washer (hole): outer radius and inner radius → A = π(R^2 − r^2).
• 5. Express radii correctly (distance from axis; subtract y-values or x-values appropriately).
• 6. Simplify integrand algebraically before integrating.
• 7. Integrate and evaluate definite integral; keep π factor.

Common Mistakes / Tips

• Always match the variable of the function to the axis you rotate around (x for x-axis, y for y-axis).
• Determine which curve gives the outer radius and which gives the inner radius for the entire interval. If curves cross within the interval, split integral at intersection points.
• When revolving about a horizontal line y = k (or vertical line x = k), radii are distances to that line: use |k − y(x)| or |k − x(y)|.
• Simplify algebra (expand and combine like terms) before integrating to reduce errors.
• Pull constants (like π) outside integrals.
• Check signs carefully when evaluating definite integrals; evaluate upper bound first, then subtract lower bound.

Summary Table (Methods and When To Use)

Action Items / Next Steps

• Practice problems: set up and evaluate volumes for regions rotated about x-axis, y-axis, and horizontal/vertical lines.
• Practice converting functions: solve y = f(x) for x = g(y) when required.
• Before integrating, always:
  • find correct bounds,
  • decide outer and inner radii,
  • simplify integrand.
• Next lecture preview: cylindrical shells method (useful when discs/washers are awkward); note variable roles swap for shells (revolve about y-axis tends to use x, and revolve about x-axis tends to use y).