key term - $r(y)$

Definition

$r(y)$ 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.

5 Must Know Facts For Your Next Test

1. The function $r(y)$ represents the radius of the cross-section of a solid at a given height $y$, which is essential for determining the volume of the solid using the disk or shell method.
2. The volume of a solid of revolution can be calculated by integrating the area of the cross-sections, where the area of each cross-section is given by $ extbackslash pi[r(y)]^2$.
3. The disk method for finding the volume of a solid of revolution involves slicing the solid into thin disks and summing the volumes of the disks, which are calculated using the function $r(y)$.
4. The shell method for finding the volume of a solid of revolution involves slicing the solid into thin cylindrical shells and summing the volumes of the shells, which are calculated using the function $r(y)$.
5. The function $r(y)$ must be a continuous and well-defined function over the interval of integration to ensure accurate volume calculations.

Review Questions

• Explain how the function $r(y)$ is used in the disk method for determining the volume of a solid of revolution.
  • In the disk method, the solid of revolution is divided into thin disks perpendicular to the axis of rotation. The volume of each disk is calculated using the formula $ extbackslash pi[r(y)]^2 extbackslash dy$, where $r(y)$ represents the radius of the cross-section at a given height $y$. By integrating the area of the disks over the height of the solid, the total volume of the solid can be determined.
• Describe the role of the function $r(y)$ in the shell method for finding the volume of a solid of revolution.
  • The shell method for finding the volume of a solid of revolution involves dividing the solid into thin cylindrical shells. The volume of each shell is calculated using the formula $2 extbackslash pi r(y) extbackslash dy extbackslash extbackslash Delta r$, where $r(y)$ represents the radius of the cross-section at a given height $y$, and $ extbackslash Delta r$ is the thickness of the shell. By integrating the volumes of the shells over the height of the solid, the total volume of the solid can be determined.
• Analyze the importance of the function $r(y)$ in the context of determining the volume of a solid by slicing, and explain how the properties of this function can affect the accuracy of the volume calculation.
  • The function $r(y)$ is crucial in the context of determining the volume of a solid by slicing because it provides the necessary information about the radius of the cross-sections at different heights. This function must be continuous and well-defined over the interval of integration to ensure accurate volume calculations. If the function $r(y)$ is not continuous or has discontinuities, the volume calculation may become more complex or even impossible using the disk or shell methods. Additionally, the properties of $r(y)$, such as its shape and behavior, can significantly impact the complexity of the volume integral and the accuracy of the final result.

Related terms

Volume of Revolution:

The volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis.

Disk Method:

A technique for finding the volume of a solid of revolution by dividing the solid into thin disks and summing the volumes of the disks.

Cylindrical Shells: A method for finding the volume of a solid of revolution by dividing the solid into thin cylindrical shells and summing the volumes of the shells.