Area of the shaded region is

The correct answer is Option (3) → \( \int_{0}^{2}x\,dx+\int_{2}^{4}(4-x)\,dx \)

The shaded region lies above the x-axis. Its upper boundary is the line \( y=x \) up to the meeting point (2, 2), and the line \( y=4-x \) beyond it.

So the area is split at \( x=2 \):

    \( \displaystyle \text{Area}=\int_{0}^{2}x\,dx+\int_{2}^{4}(4-x)\,dx \)

Its value is \( 2+2=4 \), which agrees with the area \( \frac{1}{2}\times 4\times 2 \) of the triangle.