Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = 5x2 y = 0 x = 3 (a) the y-axis 324 10 (b) the x-axis 1215n (c) the line y = 45 (d) the line x = 3
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 Solids of Revolution**
Consider the problem of finding the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines.
### Equations:
- \( y = 5x^2 \)
- \( y = 0 \)
- \( x = 3 \)
### Tasks:
(a) **Revolution about the y-axis**
The calculated volume is given by:
\[
\frac{324\pi}{10}
\]
This value is marked with a red cross, indicating it might be incorrect.
(b) **Revolution about the x-axis**
The calculated volume is given by:
\[
1215\pi
\]
This value is marked with a green check, indicating it is correct.
(c) **Revolution about the line \( y = -15 \)**
There is a blank space where the volume should be calculated, marked with a red cross, indicating it might be incorrect or not calculated yet.
(d) **Revolution about the line \( x = -3 \)**
There is a blank space where the volume should be calculated, marked with a red cross, indicating it might be incorrect or not calculated yet.
### Explanation:
Each scenario requires using the method of disks or washers to find the volume. Ensure to set up the correct integral with respect to the axis of rotation and perform the integration to find the volume accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff998db76-2065-477f-b7ac-e73b653f1166%2F94b7ca33-f757-4747-93cb-2c017c5873b9%2Fpg8bp7o_processed.jpeg&w=3840&q=75)
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