General Information

The second exam for the course will be given next Friday, November 5, as announced in the course syllabus. It will cover the material starting from immediately before the first exam through and including the material from Monday, November 1. There will be 4 or 5 problems, some possibly with several parts. The format will be similar to that of the first exam.

Topics to be Covered

1. Riemann Sums, the definite integral of a function over an interval [a,b]. Be prepared to compute a Riemann sum for a particular function and a given number of subdivisions (small, of course!) Also know how to derive the exact value of a definite integral of a function cx + d by taking the limit of the Riemann sums.
2. Total change of a function, and the Fundamental Theorem of Calculus (know the statement of this theorem).
3. Methods of integration
   1. Substitution
   2. Integration by parts
   3. Use of the table of integrals
4. Applications of integration such as computing areas, average values, volumes, etc. Be able to derive the appropriate integral formula by subdividing, approximating with a sum, and recognizing the sum as a Riemann sum for an appropriate function.

Review Session

If there is interest, I will be happy to schedule a review session outside of class time to help you get ready for the exam. I will be available any time after 4:00 pm on Wednesday, November 3.

Suggested Practice Problems

From the text:

• From Review Problems for Chapter 3: 4, 6 - 9, 12 - 14, 18, 26
• From Review Problems for Chapter 7: 1 - 7 ("numerical integration" means by computing an appropriate Riemann sum to approximate the integral)
• From Practice Integration problems at end of Chapter 7: look at a good sample of these and see if you can tell which method will apply.
• From Chapter 8, section 2: 1 - 3, 5, 12
• From Review Problems for Chapter 8: 1 - 4, 16 - 18, 21.

Sample Exam

I. The velocity of a decelerating car is measured at each of the following times, yielding a table of values:

t = time (sec)	0	2	4	6	8	10
v(t) = velocity (ft/sec)	10	9.5	8.1	5.8	4.0	1.2

1. Give an estimate for the total distance traveled by the car between t = 0 and t = 10 using a left-hand Riemann sum for the velocity function.
2. Is your result from part A less than or greater than the actual total distance traveled? Explain, including any assumptions you are making about the behavior of the velocity function.
3. At how many different times would you need to measure the velocity on this time interval to get the left-hand and right-hand sums to differ by .1?

II. Terminology.

1. What is the definition of the definite integral of a function f (x) over the interval [a,b]?
2. What does the Fundamental Theorem of Calculus say about definite integrals?

III. Compute each of the following integrals (the table of integrals from the text will be provided).

1. Show that entry II.17 in the table of integrals is correct, using integration by parts.
2. int x⁵ e^{3x^2} dx
3. int cos(2x) (sin(2x) + 4)^1/3 dx
4. int (x + 3) / (4 x² + 4x - 3) dx

IV. A solid has base equal to the triangle with corners at (0,0), (2,0), and (0,1) in the xy-plane. Its cross-sections by planes perpendicular to the x-axis are circles (with diameter extending the full width of the base at that point).

1. Find a formula for the cross-section area as a function of x.
2. Determine the volume of the solid.