 Find the area of the region. (Round your answer to three decimal places.)
f(x)	 = 	
15
x
 
x	 = 	0
y	 = 	5
y	 = 	15
bounded by the graphs of the equations

To find the area of the region bounded by the given graphs, we need to first identify the curves involved and the interval over which we are integrating.

### Given Information:
- The function: \( f(x) = \frac{15}{x} \)
- Vertical boundary: \( x = 0 \)
- Horizontal boundaries: \( y = 5 \) and \( y = 15 \)

### Step 1: Find the corresponding \( x \) values for the given \( y \) values.

We need to solve for \( x \) when \( y = 5 \) and \( y = 15 \):

\[
y = \frac{15}{x}
\]

For \( y = 5 \):

\[
5 = \frac{15}{x} \quad \Rightarrow \quad x = \frac{15}{5} = 3
\]

For \( y = 15 \):

\[
15 = \frac{15}{x} \quad \Rightarrow \quad x = \frac{15}{15} = 1
\]

### Step 2: Set up the integral to find the area.

The area \( A \) of the region can be found by integrating the function \( f(x) = \frac{15}{x} \) with respect to \( x \) from \( x = 1 \) to \( x = 3 \):

\[
A = \int_{1}^{3} \frac{15}{x} \, dx
\]

### Step 3: Compute the integral.

The integral of \( \frac{1}{x} \) is \( \ln |x| \). Thus:

\[
A = 15 \int_{1}^{3} \frac{1}{x} \, dx = 15 \left[ \ln |x| \right]_{1}^{3}
\]

\[
A = 15 \left( \ln 3 - \ln 1 \right)
\]

\[
A = 15 \left( \ln 3 - 0 \right) = 15 \ln 3
\]

### Step 4: Evaluate the logarithm and compute the area.

\[
A = 15 \cdot \ln(3)
\]

Now, let's calculate \( A \):

\[
A \approx 15 \cdot 1.0986122887 \approx 16.479 \text{ square units}
\]

So, the area of the region is approximately **16.479 square units**.

Would you like any details, or do you have any further questions? Here are 5 related questions to explore further:

1. How would the area change if the function was \( f(x) = \frac{20}{x} \)?
2. What is the significance of the natural logarithm in this context?
3. How would you find the area if the lower bound was \( y = 10 \) instead of \( y = 5 \)?
4. What happens to the area if you change the upper bound \( x = 3 \) to \( x = 5 \)?
5. Can you find the area between two curves, say \( f(x) = \frac{15}{x} \) and \( g(x) = \frac{10}{x} \)?

**Tip:** When dealing with integrals involving logarithmic functions, remember that the natural logarithm \( \ln(x) \) is particularly useful when integrating functions of the form \( \frac{1}{x} \).

Learn how to calculate the area of a region bounded by the function f(x) = 15/x and horizontal boundaries y = 5 and y = 15. Step-by-step solution with integral calculus and logarithmic functions. Approximate the area to three decimal places.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation