Mass and Center of Mass:

Find the mass and center of mass of the lamina tht occupies the region $D$ and has the given density function $\rho$.

• $D = \{(x,y): 0 \leq x \leq a, 0 \leq y \leq b \};, \rho(x,y) = 1 + x^2 + y^2$.
• $D$ is bounded by $y = x^2$ and $y = x + 2$;, $\rho(x,y) = kx$
• $D$ is bounded by the parabolas $y = x^2$ and $x = y^2$;, $\rho(x,y) = \sqrt{x}$.

Center of Mass and Moments of Inertia:

A lamina occupies the part of the disk $x^2 + y^2 \leq 1$ in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the $x$ - axis.

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse.

Find the moments of inertia $I_x,I_y,I_0$ for the lamina in the previous problem.

Applications:

A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean $\mu = 1000$, find the probability that both of the lamp's bulbs fail within 1000 hours.

Another lamp has just one bulb of the same type as above. If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1000 hours.

Applications:

When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the of the distance between them. Consider a circular city of radius 10 miles in which the population is uniformly distributed. For an uninfected individual at a fixed point $A(x_0,y_0)$, assume that the probability function is given by $$f(P) = \frac{1}{20}[20 - d(P,A)]$$ where $d(P,A)$ denotes the distance between points $P$ and $A$.

• Suppose the exposure of a person to the disease is the sum of the probabilities of catching the disease from all members of the population. Assume that the infected people are uniformly distributed throughout the city, with $k$ infected individuals per square mile. Find a double integral that represents the exposure of a person residing at $A$.
• Evaluate the integral for the case in which $A$ is the center of the city and for the case in which $A$ is located on the edge of the city. Where would you prefer to live?