Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. y = 3x2, y? = x 1 y y 0.30 0.30 0.25 0.25 0.20 0.20 0.15F 0.15 0.10F 0.10 0.05 0.05 0.05 0.10 0.15 0.20 0.25 0.30 0.05 0.10 0.15 0.20 0.25 0.30 y y 0.6 0.3 0.5- 0.2 0.1 0.1 0.4 0.3 0.05 0.1 0.15 0.20 0.25 -0.1 0.2 -0.2 0.1 0.1 -0.3 r 0.05 0.10 0.15 0.20 0.25 0.30 Find the area of the region.
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.
![### Sketch the Region Enclosed by the Given Curves
The curves given are:
\[ y = 3x^2 \]
\[ y^2 = \frac{1}{3}x \]
You need to decide whether to integrate with respect to \(x\) or \(y\). Below are the steps taken to enclose the region along with four different graphs showing approximations with rectangular strips inside the region.
### Diagrams Explanation:
1. **Top-Left and Top-Right Graphs:**
- This pair of graphs illustrates the region enclosed by the curves \(y = 3x^2\) and \(y^2 = \frac{1}{3}x\) in the first quadrant.
- A typical approximating rectangle (red in color) is drawn within the enclosed area (blue shaded region).
- **Axes:**
- The x-axis values range approximately from 0 to 0.3.
- The y-axis values range approximately from 0 to 0.3.
2. **Bottom-Left and Bottom-Right Graphs:**
- This pair of graphs shows a similar approach to visualizing the enclosed regions, and these might help to compare alternative views or confirm consistency.
- They demonstrate how an approximating rectangle (red) might look different from other points within the same bounded region (blue shaded).
- **Axes:**
- For the Bottom-Left graph, the x-axis values range from 0 to 0.3 and the y-axis values extend up to 0.6.
- For the Bottom-Right graph, the x-axis and y-axis scales are similar to the loci where the curves intersect.
### Objective:
You should integrate the area of the region enclosed by \(y = 3x^2\) and \(y^2 = \frac{1}{3}x\). A typical approximating rectangle has been shown to help visualize the method, either by vertical or horizontal slices.
### Final Step:
- **Find the area of the region:**
This can be done by setting up and evaluating the integral involving the bounds where these curves intersect. You can proceed with integrating with respect to either \(x\) or \(y\) depending on the setup.
[Provide the integration setup and solution steps here, which will involve solving for points of intersection, setting up the integral limits, and integrating the difference between the functions.]
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