The area bounded by the hyperbola `x^(2)-y^(2)=a^(2)` between the straigth lines `x=a and x=2a` is …............

Answer

To find the area bounded by the hyperbola \( x^2 - y^2 = a^2 \) between the straight lines \( x = a \) and \( x = 2a \), we will follow these steps:

### Step 1: Understand the hyperbola and the area to be calculated
The equation of the hyperbola is given by \( x^2 - y^2 = a^2 \). This hyperbola opens along the x-axis. We need to find the area between the lines \( x = a \) and \( x = 2a \).

### Step 2: Solve for \( y \)
From the hyperbola equation, we can express \( y \) in terms of \( x \):
\[ y^2 = x^2 - a^2 \implies y = \sqrt{x^2 - a^2} \]
This represents the upper branch of the hyperbola.

### Step 3: Set up the integral for the area
The area \( A \) bounded by the hyperbola and the vertical lines can be calculated using the integral:
\[ A = 2 \int_{a}^{2a} y \, dx \]
Here, we multiply by 2 because the hyperbola is symmetric about the x-axis.

### Step 4: Substitute \( y \) into the integral
Substituting \( y = \sqrt{x^2 - a^2} \) into the integral gives:
\[ A = 2 \int_{a}^{2a} \sqrt{x^2 - a^2} \, dx \]

### Step 5: Evaluate the integral
To evaluate the integral \( \int \sqrt{x^2 - a^2} \, dx \), we can use the formula:
\[ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log\left(x + \sqrt{x^2 - a^2}\right) + C \]
Now, we will evaluate this from \( a \) to \( 2a \).

### Step 6: Calculate the definite integral
Substituting the limits into the integral:
1. **Upper limit \( x = 2a \)**:
\[ \int \sqrt{(2a)^2 - a^2} \, dx = \frac{2a}{2} \sqrt{(2a)^2 - a^2} - \frac{a^2}{2} \log\left(2a + \sqrt{(2a)^2 - a^2}\right) \]
Simplifying:
\[ = a \sqrt{3a^2} - \frac{a^2}{2} \log\left(2a + a\sqrt{3}\right) = a^2\sqrt{3} - \frac{a^2}{2} \log(2 + \sqrt{3}) \]

2. **Lower limit \( x = a \)**:
\[ \int \sqrt{a^2 - a^2} \, dx = 0 \]

### Step 7: Combine the results
Thus, the area becomes:
\[ A = 2 \left( a^2\sqrt{3} - \frac{a^2}{2} \log(2 + \sqrt{3}) \right) \]
\[ = 2a^2\sqrt{3} - a^2 \log(2 + \sqrt{3}) \]

### Final Result
The area bounded by the hyperbola \( x^2 - y^2 = a^2 \) between the lines \( x = a \) and \( x = 2a \) is:
\[ A = 2a^2\sqrt{3} - a^2 \log(2 + \sqrt{3}) \]