Find the exact value of the volume of the object we obtain when rotating R about the r-axis. V Find the exact value of the volume of the object we obtain when rotating Rabout the y-axis. V
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![### Calculating Volumes of Solids of Revolution
Suppose that \( R \) is the finite region bounded by \( f(x) = \sqrt{x} \) and \( f(x) = \frac{x}{2} \).
#### Problem 1: Rotation About the x-axis
Find the exact value of the volume of the object we obtain when rotating \( R \) about the \( x \)-axis.
\[ V = \]
*(Enter the exact volume here)*
#### Problem 2: Rotation About the y-axis
Find the exact value of the volume of the object we obtain when rotating \( R \) about the \( y \)-axis.
\[ V = \]
*(Enter the exact volume here)*
### Explanation:
In this problem, we are given two functions:
1. \( f(x) = \sqrt{x} \)
2. \( f(x) = \frac{x}{2} \)
These functions define the boundaries of the region \( R \). The equations describe curves on the Cartesian plane, and we need to find the region bounded by these curves.
To find the volumes of the solids formed by rotating this region around the \( x \)-axis and the \( y \)-axis, we would use integral calculus techniques, specifically the Disk Method or Shell Method.
- For rotation about the \( x \)-axis, we might use the Disk Method.
- For rotation about the \( y \)-axis, we might use the Shell Method.
Please double-check the region and utilize the appropriate integral formulas to derive the exact volumes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ae9024-98f9-40e2-9992-1c0ab1fe204c%2Ff62bfcd1-8380-4643-a2cc-621dcdab63b5%2Feh35go_processed.png&w=3840&q=75)
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