Its time for the trip! After setting up your phone to measure your velocity on this trip, you head off for the big adventure. It all goes pretty well. . . until you forget your very favorite calculator at a rest stop. You cannot abide the thought of another person using the calculator for nefarious purposes, so you backtrack to pick it up. Fortunately, you were able to find it! But you lost some time. When you get close to NYC, you get stuck in traffic. It isnt too bad, but you crawl along at a slow pace. What is even worse is your phone runs out of power, so it stopped collecting data about 5.6 hours in. You estimate that you went 15 mph for the remainder of the trip after your phone stops.

The data you did collect from your phone is in Figure 2. This data represents the function v(t), which gives your velocity in mph in the direction toward NYC t hours into the trip. Once the data ends at 5.6 hours, assume your velocity was 15 mph for the rest of the trip.

Write a definite integral that expresses the distance in the direction of NYC you are from your starting point after T hours (since this is a function of T , we call it an accumulation function). Include a short analysis of units.

Recall that your trip is 232 miles. Write out and evaluate (based on Figure 2) a Riemann sum that gives a reasonably accurate approximation of how much time your trip took.

Write down the definite integral that represents the how much gas you used on your trip (your answer may involve the functions v, , and/or g). Include a short analysis of units.

Write out and evaluate a Riemann sum that gives a reasonably accurate approximation of how much gas you use on your trip.

Graph of v(t) 100 90 80 70 60 50 40 30 20 10 Velocity (mph) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Z 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 6 -10 5.6 5.8 + + Time (Hours) -20 -30 -40 -50 -60 -70 -80 -90 Graph of v(t) 100 90 80 70 60 50 40 30 20 10 Velocity (mph) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Z 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 6 -10 5.6 5.8 + + Time (Hours) -20 -30 -40 -50 -60 -70 -80 -90