The region bounded by f(#) = - 4a + 24 + 108, = 0, and y=0 is rotated about they axis. Find the volume of the solid of revolution. Find the exact value; write answer without decimals.
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.
![### Problem Statement:
The region bounded by \( f(x) = -4x^2 + 24x + 108 \), \( x = 0 \), and \( y = 0 \) is rotated about the y-axis. Find the volume of the solid of revolution.
### Task:
Find the exact value; write the answer without decimals.
### Diagram:
The provided 3D graphic represents the solid of revolution formed by rotating the region bounded by the curve \( f(x) = -4x^2 + 24x + 108 \), the line \( x = 0 \), and the line \( y = 0 \) about the y-axis. The resulting solid has a symmetrical structure around the y-axis, resembling a dome-like shape. The shaded region under the curve \( f(x) \) dictates the part of the plane that forms this solid upon rotation.
### Instructions:
- Utilize integral calculus concepts, specifically the method of disks or washers, to set up the integral for the volume.
- Ensure the integration limits align with the boundaries defined by the region.
- Solve the integral rigorously to find the exact volume.
- Write your final answer in a simplified, exact form without decimals.
### Answer Box:
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