Use the washer method to find the volume of the solid generated by revolving the shaded region about the x-axis. cos x V= 4x - 8x (Type an exact answer, using x as needed.)
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.
![**Title:** Calculating Volume using the Washer Method
**Objective:** Use the washer method to find the volume of the solid generated by revolving the shaded region about the x-axis.
**Diagram Explanation:**
The diagram shows a shaded region between two curves: \( y = 2\sqrt{\cos x} \) and \( y = 2 \), bounded between \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \). The region is revolved around the x-axis to generate a solid.
**Steps:**
1. **Identify the Outer and Inner Functions:**
- Outer function (top boundary): \( y = 2 \)
- Inner function (bottom boundary): \( y = 2\sqrt{\cos x} \)
2. **Boundaries:**
- Left: \( x = -\frac{\pi}{2} \)
- Right: \( x = \frac{\pi}{2} \)
3. **Apply the Washer Method Formula:**
The volume \( V \) can be found using the formula:
\[
V = \pi \int_{a}^{b} \left[ (f(x))^2 - (g(x))^2 \right] \, dx
\]
where:
- \( f(x) = 2 \) (outer radius)
- \( g(x) = 2\sqrt{\cos x} \) (inner radius)
- \( a = -\frac{\pi}{2} \), \( b = \frac{\pi}{2} \)
4. **Volume Calculation:**
The integrated expression:
\[
V = \frac{4\pi^2}{3} - 8\pi
\]
**Conclusion:**
By using the washer method, we calculated the exact volume \( V \) of the solid formed by revolving the shaded region around the x-axis. The answer is expressed in terms of \(\pi\) for precision.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0168238-e244-4a34-84ac-e572ee66fc4c%2F08f5197c-2740-48da-a6c6-42096b7fb8c7%2Fuix54q_processed.jpeg&w=3840&q=75)
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