Mathematics 136 – Calculus 2 Exam 2 – Review Sheet March 21, 2014

General Information

As announced in the course syllabus, the second midterm exam of the semester will be given Friday, March 28. The format will be similar to that of the first midterm and the exams from last semester.

• You will not be given a calculator for the exam. If you want to use one, you will be permitted to bring a TI-30 or similar non-graphing calculator. (Note: Some of you may have one of these calculators purchased for use in Chemistry courses here. That is also OK.)

• Use of cell phones, I-pods, I-pads, tablets, and any other electronic device besides a basic calculator is not allowed during the exam. Please leave such devices in your room or put them away in your backpack (and make sure cell phones are turned off).

What will be covered

The exam will cover the material since the last exam (Problem Sets 4, 5 and 6), namely the following material from sections 5.7, 5.8, 5.9, 5.10, 6.1, 6.2 of Stewart:

1. Integration by trigonometric substitution and partial fractions

2. Use of the table of integrals (a copy of relevant portions will be provided for your use on the exam)

3. Approximate integration using left- and right-hand Riemann sums, midpoint Riemann sums, trapezoidal and Simpson’s rules. Know how to compute each of these with just a calculator (for a relatively small n). Also know how to determine whether your result is an under- or overestimate for the Riemann sums and the trapezoidal rules (and if it is possible to tell that). You will want to review the work we did in clas to prepare and your work on Lab 1 for much of this.

4. Improper integrals (know how to tell what makes an integral improper, how to correctly set up the limit to determine if the integral converges, and determine the value if the integral does converge).

5. Areas between curves

6. Volumes of solids with known cross-sections and solids of revolution

Important Note: Many of the problems on this exam will require you to set up and compute an integral to find the quantity that is asked for. In addition to knowing how to set up the required integral, any of the methods of integration tested on the first exam (i.e. basic rules, u-substitution, integration by parts) and the new methods on this exam might be required to evaluate the integral. In other words, this exam is really a cumulative exam on the first two-thirds of the semester. Especially there were things you had not mastered on the first exam, you will need to begin your review for this exam by going back and looking at the material from sections 5.1 through 5.6 in Stewart. There will be a review for the exam in class on Wednesday, March 26. Review Problems Section 5.7/11, 13, 14, 15, 17, 19, 21, 23, 25, 27, 31 Section 5.8/1 - 21 odds Section 5.9/1, 3, 11, 13, 15 Section 5.10/1, 13, 15, 19, 21 (do a u-substition first), 23, 25, 29, 31, 55 Section 6.1/1, 3, 5, 7, 9, 11, 29, 37 Section 6.2/1, 3, 5, 9, 27, 29, 31, 33, 39 Sample Exam Questions This list is somewhat longer than the actual exam will be (to give you some idea of the range of different questions that might be asked). Unless otherwise directed, you may use any entry of the Table of Integrals from the text that applies. I. (A) Do you need partial fractions to compute

∫ \frac{t^{2} + 1}{t^{3} + 3t + 3} dt?

Explain, and if possible compute the integral with a simpler method. (B) Apply partial fraction decomposition to compute

∫ \frac{1}{x(x − 1)(x + 2)} dx

(C) Which trigonometric substitution would you apply to compute ∫ \frac{1}{u\sqrt{a^{2}−u^{2}}} du? What trigonometric integral do you get after making the substitution? Complete the derivation of the integral.

II. (A) Use the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 4 to estimate

the value of the integral ∫_{0}^{2} \sqrt{1 + x^{4}} dx, rounding your answers to 6 decimal places.

(B) Which of your answers in part A are overestimates and which are underestimates.

Explain how you can tell. (C) Which of the estimates in part A would you expect to have the smallest error. Explain.

(Review Lab 1 to see the idea here.)

III. For each of the following integrals, say why the integral is improper, determine if the integral converges, and if so, find its value.

A) ∫_{1}^{∞} \frac{1}{\sqrt[5]{x}} dx

B) ∫_{0}^{2} \frac{dx}{x^{2}−7x+6}

C) ∫_{0}^{∞} xe^{−3x} dx D) For which values of a is ∫_{0}^{∞} e^{ax} sin(x) dx convergent? Evaluate the integral for those

a.

IV. (A) Let R be the region in the plane bounded by y = 3 − x^{2} and the x-axis.

(1) Sketch the region R.

(2) Find the area of R.

(3) Find the volume of the solid generated by rotating R about the x-axis. (B) Let R be the region in the plane bounded by y = 3x and y = x^{2}.

(1) Sketch the region R.

(2) Find the area of R.

(3) Find the volume of the solid generated by rotating R about the x-axis.

(4) Find the volume of the solid generated by rotating R about the y-axis. (C) Let R be the region in the plane bounded by y = cos(πx), y = 1/2, x = −1/3 and

x = 1/3.

(1) Sketch the region R.

(2) Find the area of R.

(3) Find the volume of the solid generated by rotating R about the x-axis.

V. The height of a monument is 20m. The horizontal cross-section of the monument at x meters from the top is an isosceles right triangle with legs x/4 meters. Find the volume of the monument.