Find the Volume Vof the Solid obtained y rotating the Vegion bounded by the given Curves about the specified line. F3x, Y=3NX, about y=3 Vニ Sketch the Hig Yeg ion then on your own Sketh the Solid, and n typical disk or washer.
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 the Volume of a Solid of Revolution
**Problem Statement**
Find the volume \( V \) of the solid obtained by rotating the region bounded by the given curves about the specified line:
\[ y = 3x \]
\[ y = 3 \sqrt{x} \]
**Line of Rotation:** \( y = 3 \)
**Volume Formula:**
\[ V = \int_{\text{a}}^{\text{b}} [ \text{outer radius}^2 - \text{inner radius}^2 ] \, dx \]
**Instructions:**
1. **Sketch the region** bounded by the given curves.
2. Why sketch? To visualize the solid and understand the boundaries.
3. Identify and label the **outer and inner radius** functions.
4. Use integration limits from the region bounded by the curves.
**Steps:**
1. **Identify the intersection points** of the curves \( y = 3x \) and \( y = 3\sqrt{x} \):
Set \( 3x = 3\sqrt{x} \)
Solve for \( x \):
\[ x = \sqrt{x} \]
\[ x^2 = x \]
\[ x^2 - x = 0 \]
\[ x(x-1) = 0 \]
\[ x = 0 \quad \text{or} \quad x = 1 \]
2. **Integration limits:** From \( x = 0 \) to \( x = 1 \).
3. **Determine the radii:**
Outer radius, \( R(x) \):
\( R(x) = 3 - 3x \)
Inner radius, \( r(x) \):
\( r(x) = 3 - 3\sqrt{x} \)
4. **Set up the integral** for the volume \( V \):
\[ V = \int_0^1 \left[ (3 - 3\sqrt{x})^2 - (3 - 3x)^2 \right] \, dx \]
5. **Evaluate** the integral to find the volume.
By following these steps, you can find the volume \( V \) of the solid formed by rotating the given region about the line \( y = 3 \).
**Note:** To completely solve the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38aaaeb0-3205-4fdb-bd84-df06c53da4aa%2F586aa08c-d8a5-42cc-b622-7a76cbb51485%2Fj2gr8zw_processed.jpeg&w=3840&q=75)
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