2.2 Determining Volumes by Slicing

Calculating volumes of solids is a key skill in calculus. We'll look at different methods like cross-sectional integration, disk method, and washer method. These techniques help us find volumes of various shapes, from simple to complex.

Each method has its strengths. Cross-sectional integration works for irregular solids, while disk and washer methods are great for revolution solids. We'll also touch on the cylindrical shell method as an alternative approach for certain problems.

Volumes of Solids

Volumes using cross-sectional integration

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• Integrates area of cross-sections perpendicular to an axis (x, y, or z) to calculate volume of a solid
  • Select slicing axis based on solid's orientation and shape
  • Determine generic cross-section shape at a point along the axis (circle, square, triangle)
  • Express cross-section area as a function of position along the axis [object Object],[object Object], [object Object],[object Object], or [object Object],[object Object]
  • Integrate area function over the solid's interval to find volume [object Object],[object Object], $V = \int_{c}^{d} A(y) dy$, or [object Object],[object Object] (definite integral)
• Enables volume calculation for irregular solids (vase, sculpture) by breaking them into thin slices
  • Approximates volume as sum of cross-sectional areas multiplied by slice thickness
  • Increasing number of slices improves accuracy, approaching true volume in the limit

Disk method for revolution solids

• Calculates volume of a solid formed by revolving a region around a horizontal or vertical axis
  • Revolving around x-axis or y-axis creates a solid of revolution (bowl, spindle)
  • Radius of disk depends on the distance from the axis of revolution [object Object],[object Object] or [object Object],[object Object]
  • Disk area given by circle formula $A = \pi r^2$, substitute radius function
  • Integrate disk area over the revolution interval to find volume
• Volume formulas for disk method:
  • Region revolved around x-axis: $V = \pi \int_{a}^{b} [f(x)]^2 dx$
  • Region revolved around y-axis: $V = \pi \int_{c}^{d} [g(y)]^2 dy$
• Useful for solids with circular cross-sections (sphere, cone, paraboloid)

Washer method for hollow solids

• Calculates volume of a hollow solid formed by revolving a region between two curves around an axis
  • Revolving around x-axis or y-axis creates a hollow solid of revolution (pipe, shell)
  • Outer radius $R(x)$ or $R(y)$ and inner radius $r(x)$ or $r(y)$ depend on the distance from the axis
  • Washer area given by subtracting inner disk from outer disk $A = \pi(R^2 - r^2)$
  • Integrate washer area over the revolution interval to find volume
• Volume formulas for washer method:
  • Region revolved around x-axis: $V = \pi \int_{a}^{b} ([f(x)]^2 - [g(x)]^2) dx$, $f(x)$ outer curve, $g(x)$ inner curve
  • Region revolved around y-axis: $V = \pi \int_{c}^{d} ([f(y)]^2 - [g(y)]^2) dy$, $f(y)$ outer curve, $g(y)$ inner curve
• Useful for solids with hollow interiors (vase, cylindrical shell, washer)

Alternative Methods for Volumes of Revolution

• Cylindrical shell method (also known as method of shells) is an alternative technique for calculating volumes of revolution
  • Particularly useful when integrating with respect to the variable perpendicular to the axis of rotation
  • Involves integrating the surface area of thin cylindrical shells
• Volume of revolution formulas using the shell method:
  • For rotation around y-axis: $V = 2\pi \int_{a}^{b} x f(x) dx$
  • For rotation around x-axis: $V = 2\pi \int_{c}^{d} y g(y) dy$