Glossary

2.3 Volumes of Revolution: Cylindrical Shells

2 min read•june 24, 2024

Volumes of are a key application of , allowing us to calculate the volume of 3D solids formed by rotating 2D regions. We'll explore methods like disks, washers, and cylindrical shells, each suited for different scenarios.

Understanding when to use each method is crucial. We'll compare their strengths and learn how to handle complex rotations, including those around non-coordinate axes. This knowledge builds on our integration skills and expands our problem-solving toolkit.

Volumes of Revolution

Application of cylindrical shells method

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  • calculates volume of solid generated by rotating a region around a vertical or
  • Approximates volume using thin cylindrical shells
  • Volume of each shell is product of (2πrh2\pi rh) and thickness (dxdx or dydy)
    • rr is distance from , hh is height determined by bounding function(s)
  • Total volume obtained by integrating shell volumes over appropriate interval
    • Rotation around y-axis: V=∫ab2πxf(x)dxV = \int_{a}^{b} 2\pi x f(x) dx
    • Rotation around x-axis: V=∫cd2πyg(y)dyV = \int_{c}^{d} 2\pi y g(y) dy

Comparison of volume calculation methods

  • used when region bounded by axis of rotation and single function
    • Disks perpendicular to axis of rotation
    • Volume of disk: πr2dx\pi r^2 dx or πr2dy\pi r^2 dy
  • used when region bounded by two functions and axis of rotation
    • Washers perpendicular to axis of rotation
    • Volume of washer: π(R2−r2)dx\pi (R^2 - r^2) dx or π(R2−r2)dy\pi (R^2 - r^2) dy, RR and rr are outer and inner radii
  • Cylindrical shell method most appropriate when:
    • Region bounded by axis of rotation and single function, integral easier with cylindrical shells
    • Region bounded by two functions, neither intersects axis of rotation
    • Axis of rotation not a coordinate axis (x=ax = a or y=by = b)

Volumes from complex rotations

  • Region bounded by multiple functions:
    1. Identify intersection points to determine limits of integration
    2. Split region into subregions if necessary, calculate volume of each separately
    3. Add subregion volumes for total volume
  • Axis of rotation not a coordinate axis (x=ax = a or y=by = b):
    • Adjust radius in integral setup
      • Rotation around vertical line x=ax = a, radius is ∣x−a∣|x - a|
      • Rotation around horizontal line y=by = b, radius is ∣y−b∣|y - b|
    • Modify limits of integration based on intersection points of function(s) with axis of rotation
    • Evaluate integral to find volume of

Key Concepts in Volumes of Revolution

  • Integration is the fundamental process used to calculate volumes of revolution
  • Revolution refers to the rotation of a region around an axis to generate a three-dimensional solid
  • Cross-sections of the resulting solid are circular (for example, disks or washers)
  • represents an infinitesimally thin slice of the solid used in the integration process
  • Surface area of the cylindrical shell is an essential component in the cylindrical shell method

Key Terms to Review (21)

Axis of rotation: The axis of rotation is an imaginary line around which a solid figure, such as a shape or object, is rotated to create a three-dimensional volume. In the context of volumes of revolution, it serves as the reference point for determining how a two-dimensional shape generates volume when rotated about that line. The placement of the axis influences the resulting volume and shape of the solid created during the rotation process.
Cross-section: A cross-section is a two-dimensional shape created by slicing through a three-dimensional object. It provides a way to visualize and analyze the internal structure of solids, making it essential in understanding volumes, especially when using methods like slicing and revolution.
Cylindrical Shell Method: The cylindrical shell method is a technique used to calculate the volume of a solid of revolution by integrating the surface area of cylindrical shells. This method is particularly useful for finding volumes when the region being revolved is more easily described in cylindrical coordinates than in rectangular coordinates. By slicing the solid into thin cylindrical shells, one can find the volume by summing the areas of these shells as they are revolved around an axis.
Differential Element: A differential element is an infinitesimally small portion of a geometric figure used in calculus to approximate the volume or area during integration. In the context of volumes of revolution, it helps to simplify complex shapes into manageable parts, allowing for the calculation of total volume by summing these tiny components. This concept is essential in the method of cylindrical shells, where differential elements are used to represent thin cylindrical shells formed by rotating a function around an axis.
Disk method: The disk method is a technique used to determine the volume of a solid of revolution by integrating the cross-sectional area of disks perpendicular to an axis of revolution. The formula involves integrating $\pi [f(x)]^2$ or $\pi [g(y)]^2$ over a given interval.
Disk Method: The disk method is a technique used to calculate the volume of a three-dimensional solid by treating it as a series of circular disks stacked along an axis. It is a fundamental approach in the study of volumes of revolution, where a two-dimensional region is rotated around an axis to generate a three-dimensional shape.
Horizontal axis: The horizontal axis, also known as the x-axis, is a fundamental line in a Cartesian coordinate system that typically represents the independent variable in a graph. In the context of volumes of revolution using cylindrical shells, this axis plays a crucial role in determining the orientation and positioning of the region being revolved. Understanding its significance helps in setting up integrals for calculating volumes accurately.
Inner radius: The inner radius refers to the distance from the axis of rotation to the inner edge of a solid object when calculating volumes of revolution. This term is crucial in determining the volume of a solid generated by rotating a region around a specified axis, particularly when using the cylindrical shells method. Understanding the inner radius helps to accurately compute the volume by considering both inner and outer edges of the object being rotated.
Integration: Integration is a fundamental concept in calculus that represents the inverse operation of differentiation. It is used to find the area under a curve, the volume of a three-dimensional object, and other important quantities in mathematics and science.
Integration by parts: Integration by parts is a technique used to integrate the product of two functions. It is based on the product rule for differentiation and is expressed as $$ \int u \, dv = uv - \int v \, du $$.
Method of cylindrical shells.: The method of cylindrical shells is a technique for finding the volume of a solid of revolution by integrating along an axis perpendicular to the axis of rotation. It involves slicing the solid into cylindrical shells and summing their volumes.
Outer radius: The outer radius refers to the distance from the axis of rotation to the outer edge of a solid when calculating volumes of revolution using cylindrical shells. This term is crucial for determining how much 'space' the shell occupies, influencing the overall volume generated when a region is revolved around an axis. Understanding the outer radius helps in setting up the integral that computes the volume.
Revolution: A revolution is the complete rotation or circular motion of an object around a fixed axis or point. It is a fundamental concept in various fields, including mathematics, physics, and engineering, where it is used to describe the motion of objects and the calculation of related quantities such as volume and arc length.
Rotating Curve: A rotating curve refers to a curve in a two-dimensional plane that is rotated around a specified axis to create a three-dimensional solid. This process of rotation is essential for calculating volumes of revolution, where the shape generated is typically used to find the volume of solids like cylinders or spheres. The axis of rotation can significantly affect the shape and volume of the resulting solid.
Shell Volume Formula: The shell volume formula is a mathematical expression used to calculate the volume of a three-dimensional object by revolving a two-dimensional region around an axis. It is a fundamental concept in the study of volumes of revolution, particularly in the context of Cylindrical Shells.
Solid of revolution: A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional region around an axis. The volume of such solids can be calculated using integration techniques.
Solid of Revolution: A solid of revolution is a three-dimensional shape created by rotating a two-dimensional shape around an axis. This concept is essential for finding volumes and understanding geometric properties of these shapes when they are formed through rotation, often leading to practical applications in various fields such as engineering and physics.
Surface Area: Surface area is the total area that the surface of a three-dimensional object occupies. It is crucial for understanding how objects interact with their environment, such as in calculating material requirements or heat transfer. The calculation of surface area is especially relevant when analyzing shapes formed by rotation, measuring lengths of curves, and examining parametrically defined shapes.
Vertical Axis: The vertical axis, also known as the y-axis, is the imaginary line that runs vertically on a coordinate plane or graph. It represents the measurement or value in the upward and downward direction, perpendicular to the horizontal x-axis. The vertical axis is a crucial component in the context of volumes of revolution and cylindrical shells.
Washer method: The washer method is a technique used to find the volume of a solid of revolution when the solid has a hole in the middle. It involves integrating the difference between the outer radius and inner radius squared, multiplied by $\pi$.
Washer Method: The washer method is a technique used to calculate the volume of a three-dimensional object by treating it as a series of thin circular discs or washers. It is commonly employed in the context of finding the volumes of solids of revolution, where a two-dimensional shape is rotated around an axis to create a three-dimensional object.
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