Question: The area of the region bounded by the curve $y = x$ and x-axis, between $x = 0$ and $x = 2$ is:

Step 1 (Formula / Governing Law & Given Data): [1/4 Mark]

The area $A$ of a region bounded by a curve $y = f(x)$, the x-axis, and the ordinates $x = a$ and $x = b$ is given by the definite integral:

$$A = \int_{a}^{b} |f(x)| \, dx$$

Given: $f(x) = x$, $a = 0$, and $b = 2$.

Step 2 (Step-by-Step Mathematical Working): [1/2 Mark]

Substituting the given values into the integral formula:

$$A = \int_{0}^{2} x \, dx$$

Applying the power rule of integration $\int x^n \, dx = \frac{x^{n+1}}{n+1}$:

$$A = \left[ \frac{x^2}{2} \right]_{0}^{2}$$

Applying the Fundamental Theorem of Calculus (Upper limit - Lower limit):

$$A = \left( \frac{2^2}{2} \right) - \left( \frac{0^2}{2} \right)$$ $$A = \frac{4}{2} - 0 = 2$$

Step 3 (Final Concluded Answer): [1/4 Mark]

The area of the region is 2 sq. units.

Correct Option: (A)

💡 EL10 Concept Insight: Imagine a right-angled triangle with a base of 2 and a height of 2 (since $y=x$ at $x=2$); the area is simply $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2$, which is exactly what the integral calculates!