4. The integral represents the total work that is required from the force function lflKll from one point {)FU} to another point [x=b}. We know that for a constant application of force. the total work is represented by W = Force {F} * m. However since the question states this is a non-constant force. it's signaling that Force is represented by a function. which would be x} in the integral. Since force changes or can change at any moment along the bounds, you need to integrate the force fanction in order to appropriately calculate the area over the bounds. the result of the integral calculated with the bounds would be total area, in this case it's total work. If fix} is measured in newtons {N} the units for this integral would be N * distance units. in this case. it would be the distance unit for the bounds of O and b. For instance, if the bounds is is the distance in feet. it would be Newtons * foot. 5. The average value of a function f on an interval [a,b] is similar to how we interpret average value for distinct counts of items. For instance, the average score on a test if it only had two scores, 51 and 49, would be 50. However, what if you wanted to understand the average value over an 'infinite' amount of 'scores', or a continuous function. Integrating the function and dividing it by the length of the bounds does exactly that! f(average) = ba fa f(x)dx. We would want to use the average of a function for things that are truly continuous, things like temperature with respect to time, would be continuous considering temperature doesn't have discontinuous points over time. So in order to give you the average temperature of the day, we use integration to calculate the sum of the infinite amount of temperatures as the limit of dx (width of each reading or time) approaches zero and then divide by the length of the bounds. This is sort of similar to what we see with a in that you're summing up all of the individual measurements and then dividing by the total samples n. Another example would be traffic forecasting, basically by predicting demand which could help with urban planning and development. You would have to first model a function to represent traffic, could be done by plotting the amount of vehicles that pass through a section you're studying over time