Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. y = 19- x, y = x, and y = 0 Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type an exact answer.) O A. JOdy ов. The volume is - (Type an exact answer.)
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 Volume of Solids of Revolution Using Shell Method
**Problem Statement:**
Let \( R \) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \( R \) is revolved about the x-axis.
\[ y = 19 - x, \quad y = x, \quad \text{and} \quad y = 0 \]
**Task:**
Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice.
**Choose from the following options:**
**A.**
\[ \int ( \, \, \, \,) \, dy \]
OR
**B.**
\[ \int ( \, \, \, \,) \, dx \]
Fill the correct expressions in the boxes and find the volume of the solid. (Type an exact answer.)
**Answer Format:**
1. Integral set up for part **A** and **B**
2. Final volume calculation
\[ \text{The volume is } \boxed{\text{ }}. \]
**Understanding the Graphs and Diagrams:**
- The problem involves the region \( R \) defined by the lines \( y = 19 - x \), \( y = x \), and \( y = 0 \).
- The shell method requires revolving the area around the x-axis to form a solid and then calculating its volume.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8410a2c2-5b24-4e75-ab4b-85b3336a2648%2F8e244698-1440-4975-8722-2f1fb59e4a74%2Fkjmyfpi_processed.png&w=3840&q=75)
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