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(D) The area $A$ is given by the integral of the absolute difference between the two curves: $A = \int_{0}^{2\pi} |x - \sin x| \, dx$.Since $x \ge \si

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$2\pi^2$

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(D) The area $A$ is given by the integral of the absolute difference between the two curves: $A = \int_{0}^{2\pi} |x - \sin x| \, dx$.Since $x \ge \sin x$ for all $x \in [0, 2\pi]$,the absolute value can be removed: $A = \int_{0}^{2\pi} (x - \sin x) \, dx$.Evaluating the integral: $A = \left[ \frac{x^2}{2} + \cos x \right]_{0}^{2\pi}$.Substituting the limits: $A = \left( \frac{(2\pi)^2}{2} + \cos(2\pi) \right) - \left( \frac{0^2}{2} + \cos(0) \right)$.$A = (2\pi^2 + 1) - (0 + 1) = 2\pi^2$.

Area of the region bounded by the curves $y = \sin x$ and $y = x$ between the lines $x = 0$ and $x = 2\pi$ is:

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