Exercises

1. Note the steep drop in the lower curve around noon. What could have caused such behaviour in the solar radiation readings?
2. By finding the area under each of these curves, what type of information could be gained about the solar radiation? What would the differences in the three areas signify?
3. For this problem use 14 subintervals.
   1. Find the upper and lower approximating sums for the outer curve from 5:00 am to 7:00 pm. Given these values, what would you guess to be the amount of extraterrestrial solar energy from 5:00 am to 7:00 pm on June 20?
   2. Find the upper and lower approximating sums for the inner curve from 5:00 am to 7:00 pm. What conclusions can you draw by comparing the areas to the areas found in part a?
4. In this exercise, you will investigate what happens as the number of subintervals increases. This time, break each of the 14 subintervals into 4 smaller subintervals.
   1. Find the upper and lower approximating sums for the outer curve from 5:00 am to 7:00 pm.
   2. Find the upper and lower approximating sums for the inner curve from 5:00 am to 7:00 pm.
   3. Compare these new sums with those computed over the same interval in question 3. Are the results the same? Why or why not? What did you notice about the upper and lower approximating sums? How did it compare with the differences in the previous question.
5. In general, what will happen to the difference of the upper and lower approximating sums as the subintervals keep getting smaller? What result would you get if you took the limit of the upper or lower approximating areas as the time interval t approached 0?