The area $A$ of a region bounded by a curve $y = f(x)$, the x-axis, and the lines $x = a$ and $x = b$ is given by the definite integral: $$A = \int_{a}^{b} |f(x)| \, dx$$
Given the curve $y = x$, the limits $x = 0$ to $x = 2$, the area is: $$A = \int_{0}^{2} x \, dx$$
Using the power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1}$: $$A = \left[ \frac{x^2}{2} \right]_{0}^{2}$$ $$A = \left( \frac{2^2}{2} \right) - \left( \frac{0^2}{2} \right)$$ $$A = \frac{4}{2} - 0 = 2$$
Final Answer: 2 sq. units
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