To solve this problem, we need to calculate the total mass of the wire, which involves integrating the mass density function $\rho(x, y) = x^2 y$ along the semicircular curve $C$, the top half of the circle $x^2 + y^2 = 9$.

Step-by-Step Solution:

1. Parametrize the curve $C$: The semicircle can be parametrized as: $x = 3\cos(t), \quad y = 3\sin(t), \quad t \in [0, \pi].$ The curve represents the top half of the circle $x^2 + y^2 = 9$.

2. Arc length differential $ds$: The arc length differential $ds$ is given by: $ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt.$ For the parametrization: $\frac{dx}{dt} = -3\sin(t), \quad \frac{dy}{dt} = 3\cos(t).$ Therefore: $ds = \sqrt{(-3\sin(t))^2 + (3\cos(t))^2} dt = \sqrt{9\sin^2(t) + 9\cos^2(t)} dt = \sqrt{9} dt = 3dt.$

3. Substitute $(x, y)$ and $ds$: Substituting the parametrization and $ds$, the total mass $M$ is: $M = \int_C \rho(x, y) \, ds = \int_0^\pi \rho(3\cos(t), 3\sin(t)) \cdot ds.$ Since $\rho(x, y) = x^2 y$, we substitute: $\rho(3\cos(t), 3\sin(t)) = (3\cos(t))^2 \cdot (3\sin(t)) = 9\cos^2(t) \cdot 3\sin(t) = 27\cos^2(t)\sin(t).$ Also, $ds = 3dt$. Thus: $M = \int_0^\pi 27\cos^2(t)\sin(t) \cdot 3 \, dt = \int_0^\pi 81\cos^2(t)\sin(t) \, dt.$

4. Simplify and compute the integral: Use the substitution $u = \cos(t)$, so: $du = -\sin(t) dt, \quad \sin(t) dt = -du.$ When $t = 0$, $u = \cos(0) = 1$. When $t = \pi$, $u = \cos(\pi) = -1$. The integral becomes: $M = 81 \int_1^{-1} u^2 \cdot (-1) \, du = 81 \int_{-1}^1 u^2 \, du.$ Now compute the integral: $\int_{-1}^1 u^2 \, du = \left[\frac{u^3}{3}\right]_{-1}^1 = \frac{(1)^3}{3} - \frac{(-1)^3}{3} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3}.$ So: $M = 81 \cdot \frac{2}{3} = 54.$

Final Answer:

The total mass of the wire is: $\boxed{54}.$

Let me know if you want further clarification or additional help!

Related Questions to Explore:

1. How would the mass calculation change if the density function were $\rho(x, y) = x^2 + y^2$?
2. What is the length of the semicircular wire in this problem?
3. How can we adapt this method to find the mass of the full circular wire?
4. What happens if the density is constant? How does that simplify the integral?
5. How do we parametrize a quarter-circle instead of a semicircle?

Tip:

Always carefully parametrize the curve and ensure the limits of integration match the curve’s domain.