Math Problem Statement
Solution
To solve the problem, we need to:
- Graph the region bounded by , , , and .
- Compute the area enclosed between the curves.
Steps to Solve:
-
Graph the functions:
- : This is an exponential growth function.
- : This is a hyperbolic decay function.
The boundaries and are vertical lines.
-
Set up the area calculation: The area is determined by the integral of the difference where , from to .
The integral is:
-
Solve the integral:
- Compute .
- Compute .
Applying the definite integrals:
Substitute the bounds: Simplify:
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 , , and the vertical lines and . The shaded gray area represents the enclosed region.
The area, as calculated earlier, is:
Would you like me to calculate the numeric approximation for this area or explain any step in more detail?
Follow-Up Questions:
- Would you like the numerical value of the computed area?
- Do you need a breakdown of how definite integrals work for this problem?
- Should I explain how to determine which function is greater over the interval?
- Would you like a detailed explanation of how to set up similar problems with different boundaries?
- 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
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