1. Interpreting the integral.

a. Determining the distance from a velocity graph.

b. Analysis of the units of the result from an integral.

c. Interpretation of the integral as an area, if the function is always positive (or difference of areas, if the function is not always positive.)

2. Approximating the integral by left- and right-sums.

a. Actually carrying this out, by hand for small , or on a calculator or computer for any . Carrying it out for functions defined by a table of values, as in section 1, or for a function defined by a formula.

b. Understanding how to give several different approximations, based on different values of , and then how to use that information to give one value for the integral. (Sec. 2, problems 1-5.)

c. Understanding why sometimes the left- and right-sums are the same, but the approximation isn't very good yet (so you need a larger ). Being able to explain what is happening, including sketching a relevant graph. Example: for n=2.

b. What is the difference between left- and right-sums and upper-sums and lower-sums?

3. Determining the number of subintervals needed to get a certain accuracy of approximation, for the integral of a monotonic function. (formula, page 162. Graphical explanation, pages 153-155.)

(When we say that we want the final approximation of the integral accurate to one decimal place, we are assuming that the final approximation will be the average of the left-sums and right-sums for whatever you choose. In order to have that average accurate to one decimal place, we need to have the left-sum and the right-sum differ by 0.1 or less.)

• a. Interpreting the area as an integral, taking into account that some of the graphs are below the x axis.
• b. Finding the average value of a continuous function.
  • Be able to do this.
  • Be able to explain what you are doing, including sketching a relevant graph.

6. Using the Fundamental Theorem of Calculus.

• a. Carrying it out for a function whose anti-derivative you know. (Sec. 4, problem 7)
  • b. Interpreting this. (Sec. 4, problems 1, 11, 13, 14, 17, 18)

5. Approximating the limit of a function as the variable goes to infinity.