- Use a double integral to calculate the area of a region, volume under a surface, or average value of a function over a plane region
- Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region
- Solve problems involving improper double integrals
Optimizing Storage and Traffic Flow for a Small Bakery
You are helping a friend who recently opened a small bakery in your neighborhood. The bakery has a rectangular customer area, and your friend needs help with several calculations to optimize operations. The main floor area where customers walk and browse is bounded by the front counter (modeled by the line [latex]y = 2x[/latex]) and a curved display case (modeled by [latex]y = x^2[/latex]), where [latex]x[/latex] and [latex]y[/latex] are measured in meters.
Your friend wants to know the total floor area available for customer movement, the average customer density during peak hours, and how to model the probability that customers complete their orders within a certain time frame.
The customer density during the Saturday morning rush can be modeled by [latex]D(x, y) = 6xy[/latex] customers per square meter at any point [latex](x, y)[/latex] in the browsing region. Additionally, your friend has noticed that the time customers spend deciding on their order and the time spent at the register follow independent exponential distributions, which you’ll use to calculate service probabilities.