What is the area of the region bounded by the lines `y=x.y=0` and x=4?

Options

A. 4 sq. units
B. 8 sq. units
C. 12 sq. units
D. 16 sq. units

Answer

To find the area of the region bounded by the lines \( y = x \), \( y = 0 \), and \( x = 4 \), we can follow these steps:

### Step 1: Understand the Boundaries
The lines given are:
- \( y = x \): This is a diagonal line that passes through the origin and has a slope of 1.
- \( y = 0 \): This is the x-axis.
- \( x = 4 \): This is a vertical line that intersects the x-axis at the point (4, 0).

### Step 2: Sketch the Region
Draw the coordinate axes (x-axis and y-axis). Plot the line \( y = x \), which will intersect the x-axis at (0, 0) and the line \( x = 4 \) at the point (4, 4). The line \( y = 0 \) is simply the x-axis.

### Step 3: Identify the Vertices of the Bounded Region
The vertices of the triangle formed by these lines are:
- Point A (0, 0): Intersection of \( y = x \) and \( y = 0 \).
- Point B (4, 0): Intersection of \( y = 0 \) and \( x = 4 \).
- Point C (4, 4): Intersection of \( y = x \) and \( x = 4 \).

### Step 4: Calculate the Area of the Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case:
- The base of the triangle is the distance along the x-axis from (0, 0) to (4, 0), which is 4 units.
- The height of the triangle is the distance along the y-axis from (4, 0) to (4, 4), which is also 4 units.

Thus, we can substitute these values into the area formula:
\[ A = \frac{1}{2} \times 4 \times 4 = \frac{1}{2} \times 16 = 8 \text{ square units} \]

### Conclusion
The area of the region bounded by the lines \( y = x \), \( y = 0 \), and \( x = 4 \) is \( 8 \) square units.

---