Evaluation of definite integral using double-angle identity:
Note that $\sin\frac{x}{2} \cos\frac{x}{2} = \frac{1}{2}\sin x$.

Evaluation of iterated double integral:
First evaluate the inner integral with respect to $x$:
$\int_0^{\cos y} x \sin y \, dx = \sin y \left[ \frac{x^2}{2} \right]_0^{\cos y} = \frac{1}{2} \sin y \cos^2 y$.
Now evaluate the outer integral with respect to $y$:
$\int_0^\pi \frac{1}{2} \cos^2 y \sin y \, dy$.
Let $u = \cos y \implies du = -\sin y \, dy$.
When $y = 0, u = 1$; when $y = \pi, u = -1$.
$\int_1^{-1} \frac{1}{2} u^2 (-du) = \int_{-1}^1 \frac{1}{2} u^2 du = \frac{1}{2} \left[ \frac{u^3}{3} \right]_{-1}^1 = \frac{1}{6} (1 - (-1)) = \frac{2}{6} = \frac{1}{3}$.

Option breakdown:

• $\frac{1}{2}$ is an incorrect evaluation.
• $\frac{1}{3}$ is the exact value of the double integral.
• $\frac{1}{5}$ is incorrect.
• $-\frac{1}{3}$ has an incorrect negative sign.

Comparison Property of Definite Integrals (Monotonicity):
If $f$ and $g$ are integrable on $[a, b]$ with $a \le b$, and if $f(x) \ge g(x)$ for all $x \in [a, b]$, then:
$f(x) - g(x) \ge 0 \implies \int_a^b [f(x) - g(x)] dx \ge 0 \implies \int_a^b f(x) dx \ge \int_a^b g(x) dx$.
This is the fundamental monotonicity property of Riemann integration.

Option breakdown:

• They can be equal if $f(x) = g(x)$ almost everywhere.
• $\int_a^b f(x)dx \ge \int_a^b g(x)dx$ is the correct integral comparison inequality.
• The inequality is reversed.
• Integrals exist since the problem states $f$ and $g$ are integrable.

Calculation of displacement via definite integration:
Displacement $\Delta s$ over the time interval $[0, 3]$ is the definite integral of velocity $v(t)$:
$\Delta s = \int_0^3 v(t) dt = \int_0^3 (t^2 - 2t) dt$.
Integrating term by term:
$\Delta s = \left[ \frac{t^3}{3} - t^2 \right]_0^3 = \left( \frac{3^3}{3} - 3^2 \right) - 0 = (9 - 9) - 0 = 0\text{ m}$.
(Note: Total distance traveled would consider $|v(t)|$, but displacement is the net integral, which is exactly 0 m).

Option breakdown:

• $0\text{ m}$ is the exact net displacement.
• $1\text{ m}$ is incorrect.
• $\frac{1}{2}\text{ m}$ is incorrect.
• $3\text{ m}$ is incorrect.