Refer to the figure and find the volume generated by rotating the given region about the specified line. Rz about OA V = y С (), 2 B (1, 2 ) R2 y = 2 Vx, R3 R1 A (1,0)
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.
![**Transcription for Educational Website**
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**Task: Determining the Volume of a Rotated Region**
Refer to the figure below and calculate the volume \( V \) generated by rotating a specified region about a given line.
**Region to Rotate: \(\mathcal{R}_3\) about \( OA \)**
**Volume Expression:**
\[ V = \]
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**Graph Explanation:**
The figure illustrates a coordinate system with the \( x \)-axis and \( y \)-axis intersecting at the origin \( O(0,0) \). Key points on the graph are \( A(1,0) \), \( B(1,2) \), and \( C(0,2) \).
The region is divided into three distinct areas labeled:
- \(\mathcal{R}_1\): A blue-shaded region extending from \( A \) to \( B \) on the line connecting these two points.
- \(\mathcal{R}_2\): A green-shaded region above the line \( y = 2\sqrt[4]{x} \), which curves from point \( O \) through point \( B(1, 2) \).
- \(\mathcal{R}_3\): A yellow-shaded region that is the area of interest for rotation. It is bounded by the line \( y = 2\sqrt[4]{x} \), the vertical line \( x = 0 \), and horizontal line \( y = 2 \).
Please calculate \( V \), the volume formed when the region \(\mathcal{R}_3\) is revolved around the line \( OA \).
**Equation of the Curve:**
\[ y = 2 \sqrt[4]{x} \]
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Note: Ensure the calculation follows the appropriate methods for finding volumes of solids of revolution, considering the specified axis of rotation (line \( OA \)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3eb5c7f-a6b2-4d9a-9710-8653ecb65ccc%2F6180a253-4a32-4d86-9a63-7b282d5ef612%2Fsn9asrrb_processed.png&w=3840&q=75)
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