Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. y = 2(3 - x) y = 0 X = 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.
![### Volume of a Solid of Revolution
#### Problem Statement:
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.
#### Given Equations:
1. \( y = 2(3 - x) \)
2. \( y = 0 \)
3. \( x = 0 \)
### Steps for Solution:
1. **Sketch the Region:**
- The first equation \( y = 2(3 - x) \) is a linear equation. It can be rewritten as \( y = 6 - 2x \), which is a straight line with a y-intercept of 6 and a slope of -2.
- The second equation \( y = 0 \) is the x-axis.
- The third equation \( x = 0 \) is the y-axis.
The region enclosed by these equations forms a triangle with vertices at \( (0, 0) \), \( (0, 6) \), and \( (3, 0) \).
2. **Setting Up the Integral:**
Since we're revolving about the y-axis, we switch the variable to \( x \). Therefore, we solve for \( x \) from the equation \( y = 6 - 2x \):
\[
y = 6 - 2x \implies x = \frac{6 - y}{2}
\]
- The limits for \( y \) will go from 0 to 6.
3. **Volume of the Solid:**
The formula for the volume \( V \) of a solid of revolution about the y-axis is given by:
\[
V = \pi \int_{a}^{b} [f(y)]^2 \, dy
\]
In this case, \( f(y) = \frac{6 - y}{2} \).
So, the volume integral becomes:
\[
V = \pi \int_{0}^{6} \left( \frac{6 - y}{2} \right)^2 dy
\]
4. **Evaluate the Integral:**
\[
V = \pi \int_{0}^{6} \left( \frac{6 - y}{2} \right)^2 \, dy = \pi \int_{0}^{6} \frac{(6 - y)^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad118f04-c8c0-46fe-8652-72d195094665%2Fab93666d-900e-471e-aee1-aa0066cc410c%2F7sefyai_processed.jpeg&w=3840&q=75)
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