Exam II: Wed, November 3. Review session: Mon, November 1. SI Review session: Tue, November 2. Topics: FTC 5.3, Net Change Th 5.4, substitution rule 5.5, integration by parts 7.1, trigonometric integrals 7.2, trigonometric substitution 7.3, integration of rational functions 7.4, areas between curves 6.1.

Problems 1. FTC, Part 1: Differentiation of integrals. Sec. 5.3 7-18 2. Net change Th. Motion in a straight line. Given velocity function v(t), find displacement and total distance. Sec. 5.4 53-54 3. Substitution indefinite integrals: definite integrals: Sec. 5.5 1-70 4. By parts: Sec. 7.1 1-36, 63-64 5. Trigonometric integrals: Sec. 7.2 1-46 6. Trigonometric substitution: Sec. 7.3 1-30 etc.

7. Rational functions: Sec. 7.4 7-38 Completing square: Sec. 7.4 53-54 8. Area between curves Set up the area between the following curves: a) b) Sec. 6.1 5-26

General strategy for integration: some recommendations Simplify the integrand; Try obvious substitution; Classify the integrand according to its form: Trigonometric f-n  try substitution as in Sec. 7.2 (if applicable) Product of f-ns  try by parts Rational f-n  try partial fractions Radicals if if   try try trigonometric rational substitution substitution as in Sec. 7.3 4. Manipulate the integrand (e.g. algebraic, trigonometric manipulations); 5. Combine different methods; 6. Be creative!!!

Areas between curves S x y a y=f(x) b y=g(x) Consider region S between y=f(x), y=g(x), x=a, x=b, where f, g – continuous, f(x)g(x) x[a,b]. A – area of region S x y a y=f(x) b y=g(x) Divide [a,b] into n subintervals of width Approximate an area of i-th strip by a rectangle with base x and height where is some sample point in the interval [xi,xi+1]. Then

y S2 x a y=f(x) b y=g(x) S1 f(x)g(x) for some x[a,b] g(x)f(x) for some x[a,b] Then split S into sub-regions and find an area of each. Total area is S x y c x=f(y) d x=g(y) Consider region S between x=f(y), x=g(y), y=c, y=d, where f, g – continuous, f(y)g(y) y[c,d]. Then