What is the area bounded by the curve f(x) and y = 3?

Calculation:

Given,

The function is f(x) = |x - 3| , and we need to find the area bounded by the curve and the line y = 3.

To find the points of intersection, we set the function equal to 3:

\( |x - 3| = 3 \)

Solving for x :

- For\( x \geq 3 \), \(x - 3 = 3 \), which gives x = 6 .
- For ( x < 3 ), 3 - x = 3 , which gives x = 0 .

Therefore, the points of intersection are x = 0  and x = 6 .

The area can be calculated by integrating the difference between the curve and the line from x = 0 to x = 6. The integral is split into two parts due to the absolute value function:

\( A = \int_{0}^{3} (3 - x) \, dx + \int_{3}^{6} (x - 3) \, dx \)

For x in [0, 3] , ( f(x) = 3 - x ), and for ( x in [3, 6] ), ( f(x) = x - 3 ).

Compute both integrals:

- For x in [0, 3] :

\( \int_{0}^{3} (3 - x) \, dx = \left[ 3x - \frac{x^2}{2} \right]_{0}^{3} = 9 - 4.5 = 4.5 \)

- For x in [3, 6] :

\( \int_{3}^{6} (x - 3) \, dx = \left[ \frac{x^2}{2} - 3x \right]_{3}^{6} = 4.5 \)

Step 4: The total area is the sum of the two areas:

\( A = 4.5 + 4.5 = 9 \, \text{square units} \)

∴ The total area bounded by the curve and the line is 9 square units.

The correct answer is Option (4):