Math 3B, Final Exam ReviewSpring 2007The problems on the final will be similar to the ones from the homework assignments,quizzes and midterms. The exam will be about twice as long as the midterms and willemphasize the newer material covered since the last midterm (sections 6.1-6.5, 8.1-8.3).However, it will be cumulative and may include questions on any of the material coveredthis quarter (sections 4.10, 5.1 - 5.5, 7.1-7.5, 7.8). Also, be advised that to do many of theproblems on the new material, you will need to evaluate integrals and that requires knowledgeof integration techniques, etc. Below is an outline of the different topics you should know,along with lists of practice problems from Stewart. This list is not exhaustive: only themost important topics from earlier sections have been included. The midterm review sheetsprovide information on additional topics.1. Areas of Regions (6.1, 5.2). Calculate the area between two curves y = f(x) andy = g(x). Often this requires finding where the graphs intersect by solving f(x) = g(x)for x. But be careful, drawing an acurate picture of the region can be crucial. (6.1 #9, 13)2. Volumes of Revolution. If R is a region in the xy-plane, calculate the volume ofthe solid obtained when R is rotated about one of the axes or a line parallel to one ofthe axes. (6.2 # 3, 9, 25, 27, 29, 31, 35, 6.3 # 5, 9, 19)(a) Disk Method (6.2). V =Rbaπr2dx. Use this method w hen R is the regionunder the graph of y = f(x), and it is being rotated about a HORIZONTAL line.If the axis of revolution is y = c, then r = f(x) − c.(b) Washer Method (6.2). V =Rbaπ(r22− r21) dx. Use this when R is the regionbetween two curves y = f(x) and y = g(x) for a ≤ x ≤ b, and it is rotated ab outa HORIZONTAL line. If y = c is the axis of revolution, then r1= f(x) − c andr2= g(x) − c.(c) Cylindrical Shells (6.3). V =Rba2πrh dx. Use this method when R is theregion between two curves y = f(x) and y = g(x) for a ≤ x ≤ b, and it is rotatedabout a VERTICAL line. If x = c is the axis of revolution, then r = x − c andh = f(x) − g(x).(d) NOTE: If the region is between two curves with equations x = f(y) and x = g(y),then you should integrate with respect to y. In this case, if the axis of revolutionis Vertical, use Washers or Disks. If it is horizontal, use Cylindrical Shells.3. Work (6.4). Calculate the work done moving an object with a variable force F (x)from x = a to x = b. You may also need to find an equation for F (x) from a wordproblem. ((6.4 # 15, 29))4. Aver age Value (6.5). Calculate the average value of a function f(x) on an interval[a, b]. (7.1 # 59)15. Arclength (8.1). Calculate (or at least set up an integral for) the arclength of asegment of a curve y = f(x) (or x = g(y)). (8.1 # 9, 11, 19)6. Surface Area (8.2). Calculate (or at least set up integrals for) the surface area ofthe surface obtained by rotating a curve y = f(x) about the x- or y-axis. (8.2 # 3, 5,11, 26)7. Hydrostatic Force (8.3). Compute the hydrostatic force on the side of an objectsubmerged in a liquid. (8.3 # 11, 15)Earlier Topics8. Integration Techniques (7.1-7.5). Section 7.5 gives a good overview of integrationstrategies.(a) Integration Formulas. Review the table on p. 506. You need to know numbers1-10, 11, and 13; while 12, 14, 17 and 18 are also useful.(b) Substitution (5.5). Remember the two guidelines for choosing u:1) u should correspond to the “inside” function in a composition; and2) du = u0(x) dx should appear in the integrand, or at least be expressible interms of u.DON’T FORGET to either change the limits of integration OR convert the an-tiderivative back into terms of x. (5.5 23, 43, 61, 7.5 # 5, 19 )(c) Integration by Parts (7.1). Know when to use integration by parts, and howto choose u and dv (eg., u = L.I.A.T.E.). Be aware that some problems requiretwo applications of integration by parts, or some combination of u-substitutionand integration by parts. (7.1 # 7, 23, 25, 29, 31, 35)(d) Trigonometric Integrals (7.2). Be able to integrateRsinmx cosnx dx andsimple variations (like replacing the x’s with (3x) or when m = 3 and n = 1/2.)(7.2 # 1, 5, 9, 15)(e) Trigonometric Substitution (7.3). Know when to make the substitution x =a sin θ to simplify an integral, and how to convert the antiderivative back intoterms of x by drawing a right triangle. (7.3 # 7, 11, 29 (do u-sub. first))(f) Integrating Rational Functions (7.4). Know how to use long division ofpolynomials and the method of partial fractions to algebraically simplify rationalfunctions, so that they can be integrated easily. (7.4 # 8, 10, 13, 47)9. Improper Integrals (7.8). Be able to recognize an improper integral. In particular,when you see any definite integralRbaf(x) dx, you should check to see if f (x) has avertical asymptote between a and b. More importantly, you should be able to evaluatean improper integral USING LIMITS. (7.8 # 9, 13, 15, 27, 33)210. Comparison Theorem (7.8). Determine whether an improper integral converges ordiverges by using the Comparison Theorem. (7.8 # 49. 51, 53)11. Given a graph of a function f(x), you should be able to s ketch a graph of an antideriva-tive of f (x). If the antiderivative is defined by an integral, eg. F (x) =Rx0f(t) dt, thenyou should be able to compute values of F (x) by interpreting the integral as a “netarea”. (4.10 #47; 5.3 # 3.)12. You s hould b e able to write an area or a definite integral as a limit of Riemann sums,and interpret a limit of Riemann sums as a definite integral. ( 5.1 # 19; 5.2 # 17, 19,29)13. You should know what the Fundamental Theorem of Calculus, Parts I and II, say andhow to use them to (1) find the derivative of a function defined as an integral, and (2)evaluate a definite integral. (5.3 # 11, 49,