For the following exercise, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. y= x=2, and y = 3 1) ** 512n O 2) * O 3) 8n 4) 7n + 3
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.
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### Rotating Regions Bounded by Curves: Disk Method
**Objective:**
For the following exercise, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.
**Given:**
\[ y = \frac{1}{x}, \ x = 2, \ \text{and} \ y = 3 \]
**Options:**
1. \( \frac{512\pi}{3} \)
2. \( \frac{25\pi}{2} \)
3. \( 8\pi \)
4. \( 7\pi + 3 \)
**Procedure:**
1. Draw the region bounded by the curves \( y = \frac{1}{x}, \ x = 2, \ \text{and} \ y = 3 \).
2. Use the disk method to derive the volume of the region when it is rotated around the x-axis.
The disk method involves integrating the area of circular disks perpendicular to the x-axis. The general formula for the volume V of a region rotated about the x-axis is:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
where \( f(x) \) is the function defining the boundary of the region being rotated, and \([a, b]\) is the interval along the x-axis.
**Steps to Solve:**
1. Set up the integral using the given boundaries and function.
2. Compute the integral.
3. Compare the derived volume with the given options.
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