Calculus I  - Chapter 5 Review questions:

Riemann Sums, Definite integrals, rates of change, areas, and the Fundamental Theorem

Spring 2009, Prof. Janeba

1. What is the Left-hand Riemann sum for f(t) on the interval [3,5] with 4 subdivisions? (Give the formula)
2. If the graph of f is above the x-axis on [3,5] (sketch any such graph), then the sum above can be interpreted as the area of what rectangles?  (Draw 'em)
3. Continuing the previous problems, if we repeated the Riemann sum with 8, then 16, then 50, then 200 subdivisions, and so on, the (numeric) value of the sums would better and better approximate what geometric quantity? (give a description of what's being approximated)
4. What is the relationship between Riemann sums for f on [a,b] and f(x)dx (the definite integral of f over [a,b])?  (See chapter 5.2, p.300-301)  (Explain in complete sentences).
5. If the graph of f is above the x-axis on [a,b] then (the numeric value of) f(x)dx can be interpreted geometrically as ... (Explain in complete sentences.  For example, if f(x)dx = 14, what would that mean geometrically?)
6. If we interpret f(t) as the rate of change of some quantity Q, then f(t)dt can be interpreted as what in relation to the same quantity Q?  (See exam 3, question #3a, for instance, or class notes on the rate of change of grabulousness, or read
   the "Applications" section on p. 326 - 329).
7. When the graph of f is below the x-axis on [a,b] then (the numeric value of) f(x)dx can be interpreted geometrically as ... (For example, if f(x)dx = -5, what would that mean geometrically?)
8. What if the graph of f is sometimes above, sometimes below the x-axis on [a,b]?  What's the geometric interpretation of f(x)dx then?  (Draw a diagram of such a function, label appropriate bits, and explain in complete sentences.)
9. How about that Total Change concept in question #6 - does that still work if f is sometimes positive, sometimes negative on [a,b]?  (See the Jane's Bicycle Ride example from the 5/1 class notes). 
   For example, if f(t) gives an object's velocity at time t, and f(t)dt =12, what would the 12 mean, precisely?  If f was sometimes positive and sometimes negative on [a,b], what would that mean in itself?  Explain how it would affect the meaning of the "12".
10. What does the Fundamental Theorem of Calculus say, and how does it bring together Riemann Sums and antiderivatives?  Give the theorem (what the text calls "F.T.C. part 2" in 5.3), but also write several careful sentences explaining its significance - what does it do for us?
    

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