The area of the region bounded by the lines y = mx, x = 1, x = 2 and X-axis is 6 sq units, then m is equal to

Options

A. 3
B. 1
C. 2
D. 4

Explanation

To solve the problem, we need to find the value of \( m \) such that the area of the region bounded by the lines \( y = mx \), \( x = 1 \), \( x = 2 \), and the X-axis is equal to 6 square units.

### Step-by-Step Solution:

1. **Identify the Area Under the Curve**:
The area \( A \) under the line \( y = mx \) from \( x = 1 \) to \( x = 2 \) can be calculated using the definite integral:
\[ A = \int_{1}^{2} mx \, dx \]

2. **Calculate the Integral**:
We can compute the integral:
\[ A = m \int_{1}^{2} x \, dx \]
The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Therefore, we evaluate it from 1 to 2:
\[ A = m \left[ \frac{x^2}{2} \right]_{1}^{2} = m \left( \frac{2^2}{2} - \frac{1^2}{2} \right) = m \left( \frac{4}{2} - \frac{1}{2} \right) = m \left( 2 - \frac{1}{2} \right) = m \left( \frac{4 - 1}{2} \right) = m \left( \frac{3}{2} \right) \]

3. **Set the Area Equal to 6**:
According to the problem, this area is equal to 6 square units:
\[ m \left( \frac{3}{2} \right) = 6 \]

4. **Solve for \( m \)**:
To find \( m \), we can rearrange the equation:
\[ m = 6 \cdot \frac{2}{3} = 4 \]

Thus, the value of \( m \) is \( 4 \).

### Final Answer:
\[ m = 4 \]