Applications of Double Integrals

A firm manufactures smoke detectors. Two components for the detectors come from different suppliers—one in Michigan and one in Ohio. The company studies these components for their reliability and their data suggests that if $x$ is the life span in years of a randomly chosen component from the Michigan supplier and $y$ the life span in years of a randomly chosen component from the Ohio supplier, then t…

A triangular plate is bounded by the graphs of the equations $y = 2x$, $y = 4x$, and $y = 4$. The plate's density at $(x,y)$ is given by $\delta(x,y) = 4xy^2 + 1$, measured in grams per square centimeter (and $x$ and $y$ are measured in centimeters).

1. Set up an iterated integral whose value is the mass of the plate. Include a labeled sketch of the region of integration. Why did you choose the order of integration you did?

2. Determine the mass of the plate…

Let $x$ denote the time (in minutes) that a person spends waiting in a checkout line at a grocery store and $y$ the time (in minutes) that it takes to check out. Suppose the joint probability density for $x$ and $y$ is

1. What is the exact probability that a person spends between 0 to 5 minutes waiting in line, and then 0 to 5 minutes waiting to check out?

2. Set up, but do not evaluate, an iterated integral who…