Using the given information in the first picture with the graph, please explain or justify what the comparison test says about the two integrals in the second picture.Please show work so I can understand. ### Graph and Function Analysis**Graph Description:**The graph displays three continuous functions plotted over the interval from $x = 0$ to $x = 4$. The vertical axis ranges from 0 to 2. The three graphs are represented by distinct colors:- **Green Curve**: Starts at (0, 2) and decreases towards 0 as $x$ increases. It represents the function $h(x)$.- **Red Curve**: Starts below the green curve and approaches 0 as $x$ increases. This is the function $g(x)$.- **Blue Curve**: Plotted above the red curve, also decreasing as $x$ increases, representing the function $f(x)$.The graphs illustrate that $f(x) \geq g(x) \geq h(x)$ for $x > 0$.**Given Information:**- The functions $f(x)$, $g(x)$, and $h(x)$ are continuous on $(-\infty, \infty)$.- It is given that $f(x) \geq g(x) \geq h(x)$ for $x > 0$.- The integral of $g(x)$ from $0$ to $\infty$ converges, represented as $\int_0^\infty g(x) \, dx$.This information may be useful in calculus, particularly in studying improper integrals, convergence, and comparison tests. The image contains two definite integrals extending to infinity. The first integral is from 2 to infinity of a function h(x) with respect to x, represented as:\[\int_{2}^{\infty} h(x) \, dx\]The second integral is from 1 to infinity of a function F(x) with respect to x, represented as:\[\int_{1}^{\infty} F(x) \, dx\]These integrals are often used in calculus to determine the behavior and total area under a curve that extends infinitely to the right. Such integrals can converge or diverge, depending on the behavior of the functions h(x) and F(x).

From the given graph , it is clear that as x approaches ∞, the functions fx, gx, hx approach to 4…

Answered: h(x) dx 2 F(x) dx

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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Using the given information in the first picture with the graph, please explain or justify what the comparison test says about the two integrals in the second picture. Please show work so I can understand.

### Graph and Function Analysis

**Graph Description:**
The graph displays three continuous functions plotted over the interval from $x = 0$ to $x = 4$. The vertical axis ranges from 0 to 2. The three graphs are represented by distinct colors:

- **Green Curve**: Starts at (0, 2) and decreases towards 0 as $x$ increases. It represents the function $h(x)$.
- **Red Curve**: Starts below the green curve and approaches 0 as $x$ increases. This is the function $g(x)$.
- **Blue Curve**: Plotted above the red curve, also decreasing as $x$ increases, representing the function $f(x)$.

The graphs illustrate that $f(x) \geq g(x) \geq h(x)$ for $x > 0$.

**Given Information:**
- The functions $f(x)$, $g(x)$, and $h(x)$ are continuous on $(-\infty, \infty)$.
- It is given that $f(x) \geq g(x) \geq h(x)$ for $x > 0$.
- The integral of $g(x)$ from $0$ to $\infty$ converges, represented as $\int_0^\infty g(x) \, dx$.

This information may be useful in calculus, particularly in studying improper integrals, convergence, and comparison tests.

The image contains two definite integrals extending to infinity. The first integral is from 2 to infinity of a function h(x) with respect to x, represented as:

\[
\int_{2}^{\infty} h(x) \, dx
\]

The second integral is from 1 to infinity of a function F(x) with respect to x, represented as:

\[
\int_{1}^{\infty} F(x) \, dx
\]

These integrals are often used in calculus to determine the behavior and total area under a curve that extends infinitely to the right. Such integrals can converge or diverge, depending on the behavior of the functions h(x) and F(x).

Expert Solution

Step 1

From the given graph , it is clear that as $x$ approaches $\infty$ , the functions $f (x), g (x), h (x)$ approach to 4

Also, it is given that $f (x) \geq g (x) \geq h (x)$

As, $h (x) \leq g (x)$ and $\int_{1}^{\infty} g (x) d x$ converges.

So, by comparison test $\int_{1}^{\infty} h (x) d x$ converges and hence $\int_{2}^{\infty} h (x) d x$ converges

Since if we remove or odd some finite terms from infinite series it does not affect the sum of the series.

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