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➗calculus ii review

key term - $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$

Citation:

Definition

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 $[c, d]$.

5 Must Know Facts For Your Next Test

  1. The term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$ is used to calculate the volume of a solid generated by revolving a region bounded by the functions $f(y)$ and $g(y)$ around the $y$-axis.
  2. The integral in the formula calculates the difference between the volumes of the regions bounded by the functions $f(y)$ and $g(y)$ over the interval $[c, d]$.
  3. The term assumes that the region bounded by $f(y)$ and $g(y)$ is above the $y$-axis, and the volume is calculated by revolving this region around the $y$-axis.
  4. The formula can be used to find the volume of various types of solids, such as cylinders, cones, and more complex shapes.
  5. The choice of the functions $f(y)$ and $g(y)$ and the limits of integration $c$ and $d$ will depend on the specific problem being solved.

Review Questions

  • Explain the purpose and meaning of the term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$ in the context of determining volumes by slicing.
    • The term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$ is used to calculate 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 $[c, d]$. This formula is a key tool in the technique of determining volumes by slicing, where the volume of a solid is found by integrating the cross-sectional areas of thin slices along the axis of rotation.
  • Describe how the choice of the functions $f(y)$ and $g(y)$, as well as the limits of integration $c$ and $d$, affect the volume calculated using the term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$.
    • The choice of the functions $f(y)$ and $g(y)$ determines the shape of the region being revolved, and thus the overall shape of the resulting solid. The limits of integration $c$ and $d$ define the interval over which the volume is calculated. Changing any of these elements will result in a different volume calculation. For example, if the functions $f(y)$ and $g(y)$ represent a cone and a cylinder, respectively, the volume calculated using the term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$ will be the volume of the solid generated by revolving the region between the cone and the cylinder. Altering the functions or the limits of integration will change the specific volume being calculated.
  • Analyze how the term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$ relates to the overall process of determining volumes by slicing and how it can be used to solve a variety of volume problems.
    • The term $V = ext{pi} ext{int}_{c}^{d} ([f(y)]^2 - [g(y)]^2) ext{dy}$ is a key component of the process of determining volumes by slicing, as it provides a systematic way to calculate the volume of a solid generated by revolving a region around an axis. This formula can be applied to a wide range of volume problems, from simple shapes like cylinders and cones to more complex solids. By carefully selecting the functions $f(y)$ and $g(y)$ that define the region being revolved, and the appropriate limits of integration $c$ and $d$, the term can be used to find the volumes of a variety of solids. Understanding how to apply this formula and interpret the results is essential for solving volume problems using the technique of determining volumes by slicing.