Glossary
Practice Quizzes

2.2 Determining Volumes by Slicing

3 min read•june 24, 2024

Calculating volumes of solids is a key skill in calculus. We'll look at different methods like , , and . 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 as an alternative approach for certain problems.

Volumes of Solids

Volumes using cross-sectional integration

Top images from around the web for Volumes using cross-sectional integration

  • Determining Volumes by Slicing · Calculus View original

    Is this image relevant?

  • Determining Volumes by Slicing · Calculus View original

    Is this image relevant?

  • Determining Volumes by Slicing · Calculus View original

    Is this image relevant?

1 of 3

Top images from around the web for Volumes using cross-sectional integration

  • Determining Volumes by Slicing · Calculus View original

    Is this image relevant?

  • Determining Volumes by Slicing · Calculus View original

    Is this image relevant?

  • Determining Volumes by Slicing · Calculus View original

    Is this image relevant?

1 of 3
  • Integrates area of cross-sections perpendicular to an axis (x, y, or z) to calculate volume of a solid
    • Select based on solid's orientation and shape
    • Determine generic shape at a point along the axis (circle, square, triangle)
    • Express area as a function of position along the axis , , or
    • Integrate area function over the solid's interval to find volume , V=∫cdA(y)dyV = \int_{c}^{d} A(y) dy, or ()
  • 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 (bowl, spindle)
    • Radius of disk depends on the distance from the axis of revolution or
    • Disk area given by circle formula A=πr2A = \pi r^2, substitute radius function
    • Integrate disk area over the revolution interval to find volume
  • Volume formulas for :
    • Region revolved around x-axis: V=π∫ab[f(x)]2dxV = \pi \int_{a}^{b} [f(x)]^2 dx
    • Region revolved around y-axis: V=π∫cd[g(y)]2dyV = \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)R(x) or R(y)R(y) and inner radius r(x)r(x) or r(y)r(y) depend on the distance from the axis
    • Washer area given by subtracting inner disk from outer disk A=π(R2−r2)A = \pi(R^2 - r^2)
    • Integrate washer area over the revolution interval to find volume
  • Volume formulas for :
    • Region revolved around x-axis: V=π∫ab([f(x)]2−[g(x)]2)dxV = \pi \int_{a}^{b} ([f(x)]^2 - [g(x)]^2) dx, f(x)f(x) outer curve, g(x)g(x) inner curve
    • Region revolved around y-axis: V=π∫cd([f(y)]2−[g(y)]2)dyV = \pi \int_{c}^{d} ([f(y)]^2 - [g(y)]^2) dy, f(y)f(y) outer curve, g(y)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 ) 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
  • formulas using the shell method:
    • For rotation around y-axis: V=2π∫abxf(x)dxV = 2\pi \int_{a}^{b} x f(x) dx
    • For rotation around x-axis: V=2π∫cdyg(y)dyV = 2\pi \int_{c}^{d} y g(y) dy

Key Terms to Review (27)

$A(x)$: $A(x)$ is a function that represents the cross-sectional area of a solid at a particular point $x$ along its length. It is a fundamental concept in the topic of determining volumes by slicing, as it allows the calculation of the volume of a solid by integrating the area function over the interval of interest.
$A(y)$: $A(y)$ is a function that represents the cross-sectional area of a solid at a given height $y$. It is a crucial concept in the context of determining volumes by slicing, as the cross-sectional area is used to calculate the volume of the solid.
$A(z)$: $A(z)$ is a mathematical function that represents the cross-sectional area of an object at a given point $z$ along its length or height. This term is particularly important in the context of determining volumes by slicing, as the cross-sectional area is a crucial component in calculating the volume of an object using integration.
$r(x)$: $r(x)$ represents the radius of a cross-section of a solid when determining volumes by slicing. This function is crucial as it defines the shape and size of the individual slices that make up the volume of a three-dimensional object. Understanding how to compute and manipulate $r(x)$ allows for accurate volume calculations using various methods, including the disk and washer methods.
$r(y)$: $r(y)$ is a function that represents the radius of a cross-section of a three-dimensional solid at a given height $y$. This function is crucial in the context of determining the volume of a solid by slicing, as it allows the calculation of the area of each cross-section, which can then be integrated to find the total volume.
$V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$: This term represents the formula for calculating the volume of a solid generated by rotating a region bounded by the functions $f(x)$ and $g(x)$ about the $x$-axis over the interval $[a, b]$. It is a fundamental concept in the topic of determining volumes by slicing.
$V = ext{pi} ext{int}_{a}^{b} [f(x)]^2 ext{dx}$: $V = ext{pi} ext{int}_{a}^{b} [f(x)]^2 ext{dx}$ is a formula used to calculate the volume of a solid generated by rotating a function $f(x)$ around the $x$-axis over the interval $[a, b]$. This formula is the foundation for the method of determining volumes by slicing, which is a key concept in integral calculus.
$V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$: This term represents the formula for calculating the volume of a solid generated by revolving a region bounded by the functions $f(y)$ and $g(y)$ around the $y$-axis. The integral calculates the difference between the volumes of the regions bounded by the functions $f(y)$ and $g(y)$ over the interval $[c, d]$.
$V = t_{c}^{d} A(y) dy$: The expression $V = t_{c}^{d} A(y) dy$ represents the volume of a three-dimensional object that can be calculated by integrating the cross-sectional area $A(y)$ over the interval $[c, d]$. This integration process is a fundamental technique in the topic of Determining Volumes by Slicing, which allows for the calculation of the volume of various geometric shapes and solids.
$V = \\pi \\int_{c}^{d} [g(y)]^2 dy$: This formula represents the volume of a solid of revolution generated by rotating a region around the y-axis. In this expression, $g(y)$ defines the radius of the disk at a given height $y$, while the integration from $c$ to $d$ accumulates the volumes of infinitely thin disks across the specified interval. This method is essential for calculating volumes in various applications, particularly when dealing with curves and shapes that can be described mathematically.
$V = \int_{a}^{b} A(x) dx$: $V = \int_{a}^{b} A(x) dx$ is a fundamental equation used to determine the volume of a three-dimensional object by integrating the cross-sectional area function $A(x)$ over the interval $[a, b]$. This integration process allows for the calculation of the total volume enclosed within the boundaries of the object.
$V = \int_{e}^{f} A(z) dz$: $V = \int_{e}^{f} A(z) dz$ is a mathematical expression that represents the volume of a three-dimensional object. The integral calculates the volume by slicing the object into infinitesimal cross-sections and summing their areas over the given interval from $e$ to $f$. This method is particularly useful for finding the volume of irregularly shaped objects that cannot be easily calculated using basic geometric formulas.
Cross-section: A cross-section is the shape obtained by cutting a solid object with a plane. It is used to determine the volume of solids by integrating the areas of these cross-sections along an axis.
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.
Cross-Sectional Integration: Cross-sectional integration is a method used to determine the volume of an object by dividing it into thin slices or cross-sections, calculating the area of each slice, and then summing the areas to obtain the overall volume. This approach is particularly useful when the object has an irregular or complex shape that cannot be easily described by a single mathematical formula.
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.
Definite integral: The definite integral of a function between two points provides the net area under the curve from one point to the other. It is represented by the integral symbol with upper and lower limits.
Definite Integral: The definite integral represents the area under a curve on a graph over a specific interval. It is a fundamental concept in calculus that allows for the quantification of the accumulation of a quantity over a given range.
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.
Method of shells: The method of shells is a technique used to find the volume of a solid of revolution by integrating the lateral surface area of cylindrical shells. This method is particularly useful when dealing with solids formed by rotating a region around an axis, as it simplifies the process of calculating volumes by focusing on the height and radius of the cylindrical shells created during the rotation.
Slicing Axis: The slicing axis refers to the orientation or direction along which a three-dimensional object is divided or 'sliced' to determine its volume. It is a crucial concept in the context of 'Determining Volumes by Slicing', a topic that explores various methods for calculating the volume of irregularly shaped objects.
Slicing method: The slicing method is a technique used to determine the volume of a solid by integrating the cross-sectional area perpendicular to an axis. It involves summing up infinitesimally thin slices of the solid.
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.
Volume of Revolution: The volume of revolution is the three-dimensional space occupied by a two-dimensional shape when it is rotated around a fixed axis. This concept is used to calculate the volume of objects that can be generated by rotating a curve or a region in a plane about a fixed line or axis.
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.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Glossary
Glossary
2.3 Volumes of Revolution: Cylindrical Shells