Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. y = x2 - 3x, y = 4x y y y y 30- 30- 5- 25 25 4 8. 10 -10 -8 -6 –5 20 20 -5 WebAssign Plot -10 15- 15 -10 -15 10 10 -15 -20 -20 -25 -25 2. 4 8 10 -10 -8 -6 -30- -30- Find the area of the region.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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![### Enclosed Region and Area Calculation
#### Problem Statement
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) or \( y \). Draw a typical approximating rectangle.
#### Given Equations
\[ y = x^2 - 3x, \quad y = 4x \]
#### Graphs and Diagrams
1. **First Graph (Region and Approximating Rectangle - \( x \)-axis view):**
- The graph shows the region enclosed by the parabolic curve \( y = x^2 - 3x \) and the linear curve \( y = 4x \).
- The region is shaded in blue.
- A typical approximating rectangle is shown in red, illustrating how integration would be approached with respect to \( x \).
2. **Second Graph (Region and Approximating Rectangle - \( y \)-axis view):**
- The same curves are plotted: \( y = x^2 - 3x \) and \( y = 4x \) intersecting each other.
- The enclosed region is shaded blue.
- A typical approximating rectangle, drawn in red, illustrates integration with respect to \( y \).
3. **Third and Fourth Graphs (Zoomed Regions):**
- Two additional graphs focus on different portions of the \( x \)-axis. The region enclosed by the curves is clearly visible, shaded in blue.
- Similar approximating rectangles (red) are shown, indicating potential methods of integration over the specified intervals.
#### Objective
**Find the area of the region.**
An input box is provided below the graphs for the user to input the calculated area of the enclosed region.
### Calculation Steps
1. **Identify Intersection Points:**
- Determine the points where the curves intersect by solving \( x^2 - 3x = 4x \).
- Simplifying this equation will yield the intersection points.
2. **Set Up the Integral(s):**
- Depending on the chosen method (with respect to \( x \) or \( y \)), set up the integral involving the upper curve \( y = 4x \) and the lower curve \( y = x^2 - 3x \).
3. **Evaluate the Integral:**
- Compute the definite integral to find the area of the enclosed region.
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