Glossary

9.2 Volumes of Solids of Revolution

3 min read•august 7, 2024

Volumes of Solids of Revolution is all about finding the volume of 3D shapes created by spinning 2D regions around an axis. It's like making a pottery vase by spinning clay on a wheel.

We use methods like the disk, washer, and cylindrical shell to calculate these volumes. These techniques build on our integration skills, helping us solve real-world problems involving rotational symmetry.

Methods for Calculating Volumes

Disk Method

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  • Calculates the volume of a solid of revolution by approximating it with thin cylindrical disks
  • Requires integrating the area of the cross-sectional disks along the axis of rotation
  • is a circle perpendicular to the axis of rotation
    • Circle's radius is a function of the distance from the axis of rotation
  • Volume is calculated using the formula: V=∫abπ[f(x)]2dxV = \int_a^b \pi [f(x)]^2 dx (for rotation about the x-axis)
  • Useful when the cross-sections of the solid are disks or circles

Washer Method

  • Calculates the volume of a solid of revolution by approximating it with thin cylindrical washers
  • Requires integrating the area of the cross-sectional washers along the axis of rotation
  • Cross-sectional area is a washer (a ring-shaped region) perpendicular to the axis of rotation
    • Washer's outer and inner radii are functions of the distance from the axis of rotation
  • Volume is calculated using the formula: V=∫abπ([R(x)]2−[r(x)]2)dxV = \int_a^b \pi ([R(x)]^2 - [r(x)]^2) dx (for rotation about the x-axis)
    • R(x)R(x) represents the outer radius function and r(x)r(x) represents the inner radius function
  • Useful when the solid is formed by revolving a region between two curves

Volume Formula

  • General formula for calculating the volume of a solid of revolution
  • Integrates the cross-sectional area along the axis of rotation
  • Formula: V=∫abA(x)dxV = \int_a^b A(x) dx (for rotation about the x-axis) or V=∫cdA(y)dyV = \int_c^d A(y) dy (for rotation about the y-axis)
    • A(x)A(x) or A(y)A(y) represents the cross-sectional area as a function of the distance from the axis of rotation
  • Can be used with various cross-sectional shapes (disks, washers, )
  • Provides a unified approach to calculating volumes of solids of revolution

Rotating Around Axes

Axis of Rotation

  • The line around which a planar region is rotated to generate a solid of revolution
  • Can be the x-axis, y-axis, or any other parallel line
  • Choice of axis of rotation determines the method and formula used for volume calculation
  • Affects the shape and orientation of the resulting solid

Revolution about x-axis

  • Planar region is rotated around the x-axis to generate a solid of revolution
  • Cross-sections perpendicular to the x-axis are used for volume calculation
  • : V=∫abπ[f(x)]2dxV = \int_a^b \pi [f(x)]^2 dx
  • : V=∫abπ([R(x)]2−[r(x)]2)dxV = \int_a^b \pi ([R(x)]^2 - [r(x)]^2) dx
  • Cylindrical shell method: V=2π∫abxf(x)dxV = 2\pi \int_a^b x f(x) dx

Revolution about y-axis

  • Planar region is rotated around the y-axis to generate a solid of revolution
  • Cross-sections perpendicular to the y-axis are used for volume calculation
  • Disk method: V=∫cdπ[g(y)]2dyV = \int_c^d \pi [g(y)]^2 dy
  • Washer method: V=∫cdπ([R(y)]2−[r(y)]2)dyV = \int_c^d \pi ([R(y)]^2 - [r(y)]^2) dy
  • Cylindrical shell method: V=2π∫cdyg(y)dyV = 2\pi \int_c^d y g(y) dy

Geometric Components

Cross-sectional Area

  • The area of a cross-section of a solid of revolution perpendicular to the axis of rotation
  • Varies depending on the distance from the axis of rotation
  • Can be a disk (circle), washer (ring-shaped region), or cylindrical shell
  • Represented as a function of the distance from the axis of rotation (A(x)A(x) or A(y)A(y))
  • Integral of the cross-sectional area along the axis of rotation gives the volume of the solid

Cylindrical Shell

  • A thin cylindrical surface formed by rotating a narrow rectangular strip about an axis
  • Height of the shell is determined by the function being rotated
  • Radius of the shell is the distance from the axis of rotation to the rectangular strip
  • Cross-sectional area of a cylindrical shell: A(x)=2πxf(x)A(x) = 2\pi x f(x) (for rotation about the x-axis) or A(y)=2πyg(y)A(y) = 2\pi y g(y) (for rotation about the y-axis)
  • Volume of a solid using cylindrical shells: V=2π∫abxf(x)dxV = 2\pi \int_a^b x f(x) dx (for rotation about the x-axis) or V=2π∫cdyg(y)dyV = 2\pi \int_c^d y g(y) dy (for rotation about the y-axis)
  • Useful when the solid can be approximated by nested cylindrical shells

Key Terms to Review (19)

Cone: A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. In the context of volumes of solids of revolution, a cone can be formed by rotating a right triangle around one of its legs, creating a circular base and a pointed top. This shape is important for understanding how to calculate the volume of solids generated through rotation.
Cross-Sectional Area: The cross-sectional area is the area of a shape obtained by slicing a three-dimensional object with a plane. This concept is crucial in understanding the volumes of solids of revolution, as the cross-sectional area at various points along the axis of rotation allows for the calculation of the overall volume using integration.
Cylinder: A cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface at a fixed distance from the center. In the context of solids of revolution, a cylinder can be visualized as the result of revolving a rectangle around one of its sides, which serves as the axis of rotation. Understanding the properties of cylinders is essential when calculating volumes and surface areas, especially when working with different methods of integration.
Cylindrical shells: Cylindrical shells are a method used to calculate the volume of solids of revolution by visualizing the solid as a series of thin cylindrical layers. This technique involves wrapping a cylindrical shell around an axis of rotation, allowing us to express the volume in terms of the height and radius of the shell as it revolves. The method is particularly useful for integrating functions that describe curves, leading to a more straightforward calculation of the volume than other methods.
Definite Integral: A definite integral is a mathematical concept that represents the signed area under a curve between two specific points on the x-axis. It provides a way to calculate the accumulation of quantities, such as distance, area, or volume, and is fundamentally linked to the idea of antiderivatives and rates of change.
Disk method: The disk method is a technique used to find the volume of a solid of revolution by slicing the solid into thin disks perpendicular to the axis of rotation. Each disk's volume is calculated using the formula for the volume of a cylinder, and then these volumes are summed (integrated) across the entire solid. This method is particularly useful for solids formed by rotating regions bounded by curves around an axis.
Engineering: Engineering is the application of scientific and mathematical principles to design, create, and analyze structures, systems, and processes. It encompasses a wide range of disciplines and focuses on solving real-world problems through innovation and efficient resource use. This field plays a vital role in developing technology, infrastructure, and various products that improve everyday life.
Finding Bounds: Finding bounds refers to the process of determining the upper and lower limits of a quantity, especially in the context of calculating volumes of solids formed by rotating a region around an axis. This involves identifying the functions that describe the boundaries of the region being revolved, which can be crucial for using methods like the disk and washer methods. Accurate bounds help ensure that the calculated volume is correct and reflects the physical space represented by the solid.
Identifying Functions: Identifying functions involves recognizing relationships between sets of values where each input corresponds to exactly one output. This concept is foundational in mathematics, especially when analyzing graphs and equations, as it helps to determine whether a relation can be classified as a function based on the criteria of uniqueness of output for every input.
Limits of Integration: Limits of integration are specific values that define the range over which an integral is calculated. These limits are essential in determining the area under a curve or the volume of a solid, particularly when using techniques like the disk and washer methods for solids of revolution. They help establish the boundaries within which the function is analyzed, allowing for accurate computation of quantities like area, volume, or total accumulation.
Manufacturing: Manufacturing is the process of converting raw materials into finished goods through various methods, including mechanical, chemical, and electrical processes. This term connects to the idea of producing items in large quantities, where techniques like assembly lines or batch production come into play. Manufacturing plays a crucial role in economics and industry, often impacting efficiency, costs, and innovation in product development.
Pappus's Centroid Theorem: Pappus's Centroid Theorem states that the volume of a solid of revolution generated by rotating a plane figure around an external axis is equal to the product of the area of the figure and the distance traveled by its centroid during the rotation. This theorem provides a powerful method for calculating volumes and relates closely to both the shell method and the volumes of solids of revolution by emphasizing the importance of centroids in these calculations.
Representative Slice: A representative slice is a cross-section of a solid that is used to approximate the volume of the entire solid, particularly in the context of solids of revolution. This concept involves taking a thin slice of the solid at a particular position, which can be used to calculate the volume through integration. The choice of representative slice is crucial because it affects the accuracy of the volume estimation when using methods like the disk or washer method.
Sphere: A sphere is a perfectly symmetrical three-dimensional shape where all points on its surface are equidistant from its center. This constant distance, known as the radius, gives the sphere its unique round form, making it a fundamental object in geometry. Spheres can be found in various real-world applications, such as in physics and engineering, and play a crucial role when discussing volumes of solids formed through revolution.
V = π∫[a to b] (f(x))² dx: This equation represents the formula for calculating the volume of a solid of revolution when a region bounded by a curve is rotated around a horizontal axis. The integral computes the sum of infinitesimally thin cylindrical disks, each with volume given by the area of the circular face, which is determined by the square of the function value, multiplied by π. Understanding this formula is essential in visualizing how areas can be transformed into three-dimensional volumes through rotation.
V = π∫[a to b] (r² - r²) dx: This equation represents the volume of a solid of revolution formed by rotating a region around a specified axis. It specifically calculates the volume by integrating the difference between the squares of the outer radius and inner radius functions, multiplied by π. The limits 'a' and 'b' denote the interval along the x-axis where the rotation takes place, effectively capturing the full height of the solid.
Washer method: The washer method is a technique used to find the volume of a solid of revolution by integrating the area of circular cross-sections perpendicular to an axis. This method involves slicing the solid into thin disks or washers, where each washer has an inner radius and an outer radius, allowing for the calculation of the volume between two curves. The integration of these areas over a specified interval provides the total volume of the solid formed when a region in a plane is revolved around an axis.
X = g(y): In the context of finding volumes of solids of revolution, 'x = g(y)' represents a function that expresses 'x' in terms of 'y'. This is crucial when rotating a region around a horizontal axis, as it allows you to set up the integral for volume using the appropriate radius based on the function. Understanding this relationship is essential for calculating the volume of solids formed by such rotations.
Y = f(x): The equation y = f(x) represents a function where 'y' is the output value dependent on the input value 'x', which is processed through the function 'f'. This expression showcases the relationship between input and output, highlighting how changes in 'x' affect 'y'. Understanding this relationship is crucial for analyzing functions and their properties, particularly when considering transformations, limits, and integrals in calculus.
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