Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.
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 x-axis.
The equations and boundaries are:
\[ y = \frac{1}{\sqrt{5x + 6}} \]
\[ y = 0 \]
\[ x = 0 \]
\[ x = 9 \]
**Step-by-Step Explanation:**
1. **Identify the Bounded Region:**
- The curve given is \( y = \frac{1}{\sqrt{5x + 6}} \).
- The line \( y = 0 \) represents the x-axis.
- The vertical lines \( x = 0 \) and \( x = 9 \) act as the left and right boundaries, respectively.
2. **Revolving Around the x-axis:**
- To find the volume of the solid generated by revolving this bounded region about the x-axis, we use the disk method.
3. **Volume Using Disk Method:**
- The formula for the volume \( V \) of the solid of revolution using the disk method is:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
- Here, \( f(x) = \frac{1}{\sqrt{5x + 6}} \), \( a = 0 \), and \( b = 9 \).
4. **Set Up the Integral:**
\[ V = \pi \int_{0}^{9} \left( \frac{1}{\sqrt{5x + 6}} \right)^2 \, dx \]
5. **Simplify the Integrand:**
\[ \left( \frac{1}{\sqrt{5x + 6}} \right)^2 = \frac{1}{5x + 6} \]
- Now our integral becomes:
\[ V = \pi \int_{0}^{9} \frac{1}{5x + 6} \, dx \]
6. **Integral Calculation:**
- To integrate \( \frac{1}{5x + 6} \), we can use the natural logarithm (ln) rule. The integral of \( \frac{1}{u} \) is \( \ln{|u|} \), where \( u](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad118f04-c8c0-46fe-8652-72d195094665%2Fe24a2226-d2e2-4d9f-abfb-9f39656600f2%2F7aw83ui_processed.jpeg&w=3840&q=75)
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