To solve the problem, let's break it down into steps based on the given image:

1. Understand the problem:

   • We have a sphere defined by the equation:
     $P: (x-1)^2 + (y-2)^2 + (z+3)^2 = 9,$ with the constraint $x \geq 1$.
   • The density function is given as:
     $\rho(x, y, z) = \frac{x - 1}{3\pi}.$

2. Objective:

   • Find the mass of the material surface $P$ using the formula:
     $M = \int_P \rho(x, y, z) \, dS,$ where $dS$ is the surface element of $P$.

3. Approach:

   • The given sphere has a center at $(1, 2, -3)$ and a radius of 3.
   • The surface integral of the density will be restricted to the part of the sphere where $x \geq 1$.
   • Use the spherical parametrization of the sphere to calculate the surface integral.

Let me proceed to compute the integral. sympy import symbols, sin, cos, pi, integrate, sqrt

Define spherical coordinates and density function

theta, phi = symbols('theta phi') # theta: azimuthal angle, phi: polar angle x = 1 + 3*sin(phi)cos(theta) # Parametric equation for x in spherical coordinates y = 2 + 3sin(phi)sin(theta) # Parametric equation for y in spherical coordinates z = -3 + 3cos(phi) # Parametric equation for z in spherical coordinates

Density function

density = (x - 1) / (3*pi)

Compute the surface element (Jacobian factor for the sphere of radius 3)

R = 3 dS = R**2 * sin(phi)

Mass integral over the hemisphere (restricting to x >= 1 means theta ranges from 0 to pi/2)

mass_integral = integrate( integrate(density * dS, (theta, 0, pi/2)), # Azimuthal angle from 0 to pi/2 (x >= 1) (phi, 0, pi) # Polar angle from 0 to pi )

mass_integral.evalf()The mass of the material surface $P$ is calculated as:

$M = 4.5$

Let me know if you need detailed steps or further clarification!

Here are 5 related questions you might consider:

1. What are the steps for calculating surface integrals in spherical coordinates?
2. How do constraints (like $x \geq 1$) affect the limits of integration?
3. Why is the surface element $dS$ proportional to $R^2 \sin(\phi)$?
4. What changes if the density function depends on $y$ or $z$ instead of $x$?
5. How do we parametrize surfaces for other shapes, like ellipsoids?

Tip: When working with integrals over spherical surfaces, always ensure the limits of integration match the constraints of the problem!