### Understanding the Functions and the Bounded RegionConsider the following equations:\[ f(x) = x^5 + 1 \]\[ g(x) = x + 1 \]These equations define two functions. The first function, $ f(x) $, is a fifth-degree polynomial, and the second function, $ g(x) $, is a linear equation.**Objective:** Sketch the region bounded by the graphs of these equations.To solve this problem:1. **Graph $ f(x) $ and $ g(x) $:** - The graph of $ f(x) = x^5 + 1 $ will be a polynomial curve with the term $ x^5 $ indicating a steep increase/decrease as $ x $ moves away from zero. The graph will shift up by 1 unit due to the constant $ +1 $. - The graph of $ g(x) = x + 1 $ will be a straight line with a slope of 1 and a y-intercept at $ y = 1 $. This line will move upwards diagonally.2. **Find the points of intersection:** - Set $ f(x) = g(x) $: \[ x^5 + 1 = x + 1 \implies x^5 = x \] - Solve for $ x $: \[ x^5 - x = 0 \] \[ x(x^4 - 1) = 0 \] \[ x(x^2 - 1)(x^2 + 1) = 0 \] \[ x(x - 1)(x + 1)(x^2 + 1) = 0 \] - The real solutions are $ x = 0, x = 1, x = -1 $.3. **Sketch the Graphs:** - Plot the points of intersection on the graph. Ensure $ x = -1, 0, 1 $ are marked. - Draw the curve of the polynomial $ f(x) = x^5 + 1 $. - Draw the line $ g(x) = x + 1 $.**Region Bounded:** The area between the curve $ f(x) $ and the line $ g(x) $ within the interval where they intersect will be the bounded region. You will see distinct bounded ### Calculating the Area Between Curves In this lesson, we will explore finding the area of the region enclosed between two curves. The figure below shows two curves and the shaded area between them:![Graph Showing the Area Between Curves](your-image-url)#### Explanation of the GraphThe graph displays two functions intersecting at two points, forming an enclosed region. **Axes:**- The x-axis ranges from -3 to 3.- The y-axis ranges from 0 to 7.**Curves:**- There are two distinct curves shown: - A steeper curve that increases rapidly after intersecting with the second line around x = -1. - A straight line passing through the origin and inclined with respect to both axes.**Shaded Region:**- The enclosed area between these two curves is shaded light pink.- The region of interest lies between their points of intersection around x = -1 and x = 1.### Problem StatementYou are required to find the area of the shaded region. #### Steps to Solve:1. Identify the functions of the two curves.2. Determine the points of intersection.3. Set up the integral with the limits being the points of intersection.4. Integrate the difference between the two functions over the interval.```plaintextFind the area of the region.```Please enter your result in the box below:[Input Box]Using calculus and integration techniques, you can determine the exact area of the region enclosed between these curves. Ensure your calculations are accurate and reflect the correct bounds and functions.

Given that and and their graph is also given. Also the…

Answered: Find the area of the region.

Math

Advanced Math

Find the area of the region.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Understanding the Functions and the Bounded Region

Consider the following equations:

\[ f(x) = x^5 + 1 \]
\[ g(x) = x + 1 \]

These equations define two functions. The first function, $ f(x) $, is a fifth-degree polynomial, and the second function, $ g(x) $, is a linear equation.

**Objective:** Sketch the region bounded by the graphs of these equations.

To solve this problem:

1. **Graph $ f(x) $ and $ g(x) $:**
- The graph of $ f(x) = x^5 + 1 $ will be a polynomial curve with the term $ x^5 $ indicating a steep increase/decrease as $ x $ moves away from zero. The graph will shift up by 1 unit due to the constant $ +1 $.
- The graph of $ g(x) = x + 1 $ will be a straight line with a slope of 1 and a y-intercept at $ y = 1 $. This line will move upwards diagonally.

2. **Find the points of intersection:**
- Set $ f(x) = g(x) $:
\[ x^5 + 1 = x + 1 \implies x^5 = x \]
- Solve for $ x $:
\[ x^5 - x = 0 \]
\[ x(x^4 - 1) = 0 \]
\[ x(x^2 - 1)(x^2 + 1) = 0 \]
\[ x(x - 1)(x + 1)(x^2 + 1) = 0 \]
- The real solutions are $ x = 0, x = 1, x = -1 $.

3. **Sketch the Graphs:**
- Plot the points of intersection on the graph. Ensure $ x = -1, 0, 1 $ are marked.
- Draw the curve of the polynomial $ f(x) = x^5 + 1 $.
- Draw the line $ g(x) = x + 1 $.

**Region Bounded:** The area between the curve $ f(x) $ and the line $ g(x) $ within the interval where they intersect will be the bounded region. You will see distinct bounded

### Calculating the Area Between Curves

In this lesson, we will explore finding the area of the region enclosed between two curves. The figure below shows two curves and the shaded area between them:

![Graph Showing the Area Between Curves](your-image-url)

#### Explanation of the Graph

The graph displays two functions intersecting at two points, forming an enclosed region.

**Axes:**
- The x-axis ranges from -3 to 3.
- The y-axis ranges from 0 to 7.

**Curves:**
- There are two distinct curves shown:
- A steeper curve that increases rapidly after intersecting with the second line around x = -1.
- A straight line passing through the origin and inclined with respect to both axes.

**Shaded Region:**
- The enclosed area between these two curves is shaded light pink.
- The region of interest lies between their points of intersection around x = -1 and x = 1.

### Problem Statement

You are required to find the area of the shaded region.

#### Steps to Solve:
1. Identify the functions of the two curves.
2. Determine the points of intersection.
3. Set up the integral with the limits being the points of intersection.
4. Integrate the difference between the two functions over the interval.

```plaintext
Find the area of the region.
```

Please enter your result in the box below:

[Input Box]

Using calculus and integration techniques, you can determine the exact area of the region enclosed between these curves. Ensure your calculations are accurate and reflect the correct bounds and functions.

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