Your task is to estimate how far an object traveled during the time interval but

Question

Your task is to estimate how far an object traveled during the time interval but you only have the following data about the velocity of the object. To get an idea of what the velocity function might look like, you pick up a black pen, plot the data points, and connect them by curves. Your sketch looks something like the black curve in the graph below. You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue pen and draw rectangles whose height is determined by the velocity measurement at the left endpoint of each one-second interval. By using the left endpoint Riemann sum as an approximation, you are assuming that the actual velocity is approximately constant on each one-second interval (or, equivalently, that the actual acceleration is approximately zero on each one-second interval), and that the velocity and acceleration have discontinuous jumps every second. This assumption is probably incorrect because it is likely that the velocity and acceleration change continuously over time. However, you decide to use this approximation anyway since it seems like a reasonable approximation to the actual velocity given the limited amount of data. Using the left endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units. Total displacement = Total distance traveled =

Explanation / Answer

We need to find the areas of all the rectangles using the left endpoints and then sum them up. For displacement, we also have to take into account when the distance goes negative so its basically the "net distance travelled", whereas for total distance we want to take the absolute value of any negative quantity when we are adding. The width of each rectangle is 1 sec, and the height is given by the table in m/sec. So when we find the area we are doing (sec)*(m/sec) = m, so in fact we are finding the total distance. Total Displacement: We can factor out the 1 sec since all the rectangles have the same width: D = 1(4 + 3 + 1 + 3 + 1 + (-3) + (-1) + (-4)) = 1(4) = 4 meters Total Distance: Now we do the same except make all negative quantities positive: Distance = 1(4 + 3 + 1 + 3 + 1 + 3 + 1 + 4) = 1(20) = 20 meters

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