Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Find two or three classmates and a few feet of chalkboard Introduce yourself to your new friends and write all of your names at the top of the chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises some are very hard Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own are from the mathematical folklore or are independently marked Here s a hint drawing pictures e g sketching graphs of functions will always make the problem easier Trigonometric Integrals The most important rule to know for trigonometric integrals is the Pythagorean identity cos2 x sin2 x 1 This lets us translate between squares of cosines and squares of sines This is helpful for finding 7 x we can break u substitutions since cos0 sin and sin0 cos For example to cos R integrate R off of sin Rand then substitute cos7 x dx cos2 x 3 cos x dx R in terms R a cos 2and3write the rest 2 3 1 sin x cos x dx 1 u du u6 3u4 3u2 1 du u7 7 3u5 5 u3 u C sin7 x 7 3 sin5 x 5 sin3 x sin x C This trick turns the integral of cosn x sinm x into the integral of a polynomial provided that n and m are non negative and at least one of n and m are odd When they can be negative but still at least one is odd we get rational functions which we will learn how to integrate in section 7 4 By dividing the Pythagorean identity by sin2 or cos2 we get two more versions of the rule 1 tan2 x sec2 x and 1 cot2 x csc2 x Since tan0 sec2 and sec0 tan sec we can integrate tann secm if m is even or n is odd It s similar for cot and csc Sometimes though these aren t enough Then it s important the remember the double angle formulas sin2 x 1 1 cos 2x 2 cos2 x 1 1 cos 2x 2 sin x cos x The product to sum formulas are also occasionally helpful 2 sin A sin B cos A B cos A B 2 sin A cos B sin A B sin A B 2 cos A cos B cos A B cos A B 1 1 sin 2x 2 1 Evaluate the integrals Z a 6 Z 3 sin x cos x dx 2 5 cos x dx b Z c 0 Z d Z g Z x cos2 x dx 3 e 5 tan 2x sec 2x dx Z j 4 6 csc x cot x dx Z Z h k sin x cos x 3 dx n 6 tan ay dy Z 3 2 m cos cos5 sin d cos x sin x dx sin 2x Z cos 3x sin 2x dx Z f sin3 x dx x cot5 sin4 d Z sin d cos3 Z dx l cos x 1 Z o cos 2x 1 cos 4x 2 sin x dx i x 0 R 2 Evaluate sin x cos x dx in four different ways a by substituting u cos x b by substituting u sin x c by using the double angle formula for sin 2x d by integrating by parts Explain the different appearances of the answers 3 Let a be a number such that 0 a 2 Compute the volume obtained by rotating the region bounded by the curves y tan x y 0 x a about the x axis Your answer should be a function of a R 2 2 1 4 4 Find the average value of sin2 x 2 0 sin x dx Find the average values of sin x and 2 2 sin x cos x 5 Let m and n be positive integers Prove that Z a sin mx cos nx dx 0 Z 0 b sin mx sin nx dx Z 0 cos mx cos nx dx c if m 6 n if m n if m 6 n if m n R 2 6 a Use integration by parts to find a reduction formula for x 0 cosn x dx R 2 b Let n 2k 1 be odd Make a substitution to turn x 0 cosn x dx into a polynomial integral For any particular value of k you could expand this out and integrate Instead find a reduction formula for this integral c When n 2k 1 is odd solve the reduction formulas from parts a and b to find the R 2 R 2 value of x 0 cosn x dx Hint what is x 0 cos x dx d When n 2k is even the method in part b doesn t work directly and using doubleangle formulas would be extremely messy Solve the reduction formula from part a to R 2 evaluate the integral Hint what is x 0 dx 2
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