$r(y)$ 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.
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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.
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$.
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 $r(y)$.
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 $r(y)$.
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.
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.