Final answer:
To find the average value of a function over a rectangle, integrate the function over the area of the rectangle then divide by the area. This is similar to finding an area under a curve, and can be applied in various contexts such as physics for displacement, or economics for total revenue and costs.
Step-by-step explanation:
To find the average value of a function over a rectangle, we can use the concept of an integral in calculus. Consider a rectangle within a coordinate system where the function values form a curve above the rectangle. If you're given the function's equation and the rectangle's dimensions, you would integrate the function across the interval defined by the rectangle's width while taking into account the length or the height of the rectangle as well. This is analogous to finding the area under a curve, where you're essentially summing all the instantaneous values the function takes within the rectangle and then dividing by the rectangle's area to find the average value.
The formula to be used is: \(Average Value = \frac{1}{Area} \int \int_R f(x,y) dA\) where R represents the rectangle and dA is the differential area element.
For example, if you have a constant force Fave applied over a time interval At, the average value is the area of the rectangle where the force is constant. In probability, you can find the average value of a distribution by integrating the product of the probability density function and the variable over the interval. To calculate total revenue and total costs in economics, you can use the height of the respective curves at a certain quantity and multiply by the base of the rectangle, where the base is the quantity produced or sold.
When dealing with physical displacement in physics, the final displacement can be calculated by adding the areas of shapes like rectangles and triangles found under the velocity-time graph, with the average value being the displacement over the time interval.