Math Problem Statement

Graph the region bounded by f(x) = e^x, g(x) = 1/x, x = 1, and x = 3; and find its area.

Solution

To solve the problem, we need to:

  1. Graph the region bounded by f(x)=exf(x) = e^x, g(x)=1xg(x) = \frac{1}{x}, x=1x = 1, and x=3x = 3.
  2. Compute the area enclosed between the curves.

Steps to Solve:

  1. Graph the functions:

    • f(x)=exf(x) = e^x: This is an exponential growth function.
    • g(x)=1xg(x) = \frac{1}{x}: This is a hyperbolic decay function.

    The boundaries x=1x = 1 and x=3x = 3 are vertical lines.

  2. Set up the area calculation: The area is determined by the integral of the difference g(x)f(x)g(x) - f(x) where g(x)>f(x)g(x) > f(x), from x=1x = 1 to x=3x = 3.

    The integral is: Area=13(1xex)dx\text{Area} = \int_{1}^{3} \left( \frac{1}{x} - e^x \right) \, dx

  3. Solve the integral:

    • Compute 1xdx=lnx\int \frac{1}{x} dx = \ln|x|.
    • Compute exdx=ex\int e^x dx = e^x.

    Applying the definite integrals: Area=[ln(x)ex]13\text{Area} = \left[ \ln(x) - e^x \right]_{1}^{3}

    Substitute the bounds: Area=(ln(3)e3)(ln(1)e1)\text{Area} = \left( \ln(3) - e^3 \right) - \left( \ln(1) - e^1 \right) Simplify: Area=ln(3)e3(0e1)\text{Area} = \ln(3) - e^3 - (0 - e^1) Area=ln(3)e3+e\text{Area} = \ln(3) - e^3 + e

Let me now plot the region and confirm the setup visually. numpy as np import matplotlib.pyplot as plt

Define the functions

x = np.linspace(1, 3, 500) f_x = np.exp(x) g_x = 1 / x

Plot the region

plt.figure(figsize=(8, 6)) plt.plot(x, f_x, label=r"$f(x) = e^x$", color='blue') plt.plot(x, g_x, label=r"$g(x) = \frac{1}{x}$", color='orange') plt.fill_between(x, g_x, f_x, where=(g_x > f_x), color='gray', alpha=0.3, label="Enclosed Area") plt.axvline(1, color='red', linestyle='--', label="x = 1") plt.axvline(3, color='green', linestyle='--', label="x = 3")

Add labels, legend, and grid

plt.title("Region Bounded by $f(x)$, $g(x)$, $x=1$, and $x=3$") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.grid() plt.show()The graph above shows the region bounded by f(x)=exf(x) = e^x, g(x)=1xg(x) = \frac{1}{x}, and the vertical lines x=1x = 1 and x=3x = 3. The shaded gray area represents the enclosed region.

The area, as calculated earlier, is: Area=ln(3)e3+e\text{Area} = \ln(3) - e^3 + e

Would you like me to calculate the numeric approximation for this area or explain any step in more detail?

Follow-Up Questions:

  1. Would you like the numerical value of the computed area?
  2. Do you need a breakdown of how definite integrals work for this problem?
  3. Should I explain how to determine which function is greater over the interval?
  4. Would you like a detailed explanation of how to set up similar problems with different boundaries?
  5. Do you want help practicing integrals involving exponential and logarithmic functions?

Tip: When dealing with bounded areas, always carefully identify the regions where one function dominates over the other. This avoids errors in integral setup.

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Math Problem Analysis

Mathematical Concepts

Graphing functions
Definite integrals
Exponential functions
Logarithmic functions

Formulas

Area = ∫[1, 3] (g(x) - f(x)) dx
Integral of 1/x = ln|x|
Integral of e^x = e^x

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12