Math 1B PDP Worksheet Rob Bayer February 5 2009 Instructions 1 Introduce yourselves 2 Find some blackboard space a piece of chalk and decide who will be your first scribe 3 Do the problems below having a different person be the scribe for each one Improper Integrals 1 Which of the following integrals are improper For those that are improper show how you would break them up into the limit of proper integrals and then decide whether they converge or diverge Do not use the comparison test 2 Z a e sec xdx 0 0 3 Z 1 Z b Z 3 1 dx x2 1 Z4 d 1 Z f 1 e y 2 dy c dy 4y 1 1 dx x3 ex dx 1 ex Z g sin tdt 0 Z 1 2 Consider the integral 1 1 x4 3 a Explain why this integral is improper and show it diverges b What would you get if you forgot that it was improper and just evaluated its anti derivative at the endpoints Why does this answer not make any sense Z t Z 0 3 Why we do lim lim t s 0 a Consider the integral R s 2xdx Show that this integral diverges Z b Evaluate each of the following and explain why from a naive standpoint they seem like Z t i lim t 2xdx t Z t 1 ii lim t 2xdx t Z t2 1 2xdx iii lim t t Z c Convince yourself that part b really does show that Z 0 t lim 4 Find Z t is not a good definition t ln x dx 1 x2 Integration Practice 1 Find each of the following integrals Be sure to work as a group so everyone knows how to do all these problems Z Z x 1 a ex e dx c dx x 1 x Z b sec2 sin d sec Z d ln x 1 dx x2 Z t3 1 dt 3 2 Z t t cos4 t sin4 tdt Z cos3 2x sin 2xdx Z dt et Z 1 dx x x2 4 Z e f g h i j Z k Z l Z m 1 dx x x 4 x x2 4 4 dx 1 x 3x ln sec sec2 d Extra Problems If you finish early take a stab at these 1 The Weirstrass Substitution It turns out that any rational function of sin and cos and hence of sec tan csc cot etc can be turned into an ordinary rational function via the substitution t tan x2 Let s explore why x 1 t and that sin Hint right triangles 2 2 2 2 t 1 t 1 1 t2 2t b Use trig identities to show that cos x and that sin x 1 t2 1 t2 2 c Show that dx 1 t 2 dt Z Z dx d Use parts b and c to find and sec3 xdx 3 sin x 4 cos x a Show that cos x 2 Find each of the following While you could use the technique of the previous problem for a and b you should try to find a more straightforward way Z Z Z cos x 1 sin x cos x dx a dx b dx c cos x 1 sin 2x x 3x 3 Let S be a sphere of radius 1 Now take a plane a distance d 1 away from the center and slice the sphere into Rb two pieces What is the volume of each piece Hint the formula for the volume of a revolved solid is a f x 2
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