Area bounded by the curves `x = 1, x= 3, xy = 1` and x-axis is

Options

A. log 2
B. log 3
C. log 4
D. None of these

Answer

To find the area bounded by the curves \( x = 1 \), \( x = 3 \), \( xy = 1 \), and the x-axis, we can follow these steps:

### Step 1: Identify the curves and their equations
The curves given are:
- \( x = 1 \) (a vertical line)
- \( x = 3 \) (another vertical line)
- \( xy = 1 \) (which can be rewritten as \( y = \frac{1}{x} \))
- The x-axis (which is \( y = 0 \))

### Step 2: Set up the area integral
The area \( A \) bounded by these curves can be found using the integral:
\[ A = \int_{a}^{b} y \, dx \]
where \( y \) is expressed in terms of \( x \). Here, the limits of integration are from \( x = 1 \) to \( x = 3 \), and \( y = \frac{1}{x} \).

### Step 3: Write the integral
Thus, the area can be expressed as:
\[ A = \int_{1}^{3} \frac{1}{x} \, dx \]

### Step 4: Calculate the integral
The integral of \( \frac{1}{x} \) is:
\[ \int \frac{1}{x} \, dx = \log |x| + C \]
Now, we evaluate the definite integral:
\[ A = \left[ \log x \right]_{1}^{3} \]

### Step 5: Evaluate the limits
Now we substitute the limits into the integral:
\[ A = \log(3) - \log(1) \]
Since \( \log(1) = 0 \):
\[ A = \log(3) - 0 = \log(3) \]

### Final Answer
The area bounded by the curves is:
\[ \boxed{\log(3)} \]