Calculus II PLTL Fall 2014 Worksheet 6 These problems are to be done without the use of a calculator unless otherwise specified R 1 Scribe Use the substitution x 3 sin to evaluate dx x2 9 x2 R6 2 Pairs a Approximate the integral 0 1 x2 dx using the following methods i Trapezoidal Rule with n 4 ii Simpson s Rule with n 4 R6 b What is the exact value of I 0 1 x2 dx c How does the exact value I compare to the approximations in a Does the relationship between S4 and I surprise you Look at how the error bound for S4 is calculated and explain the relationship For what types of functions will this relationship hold d Why is the relationship between T4 and I not as strong Can you think of any types of functions for which the error bound for Tn will always be zero e What is the smallest value of n such that Tn I 0 001 3 Round Robin Consider the integral R0 x 5 x2 4x 5 dx a Explain what an improper integral is in your own words and discuss the approaches you can take when evaluating such an integral b Is the integral above improper Why why not c How is the integral above related to the area under the curve y the interval 5 0 d If possible determine the value of the integral x x2 4x 5 on 4 Scribe If the infinite curve y e x x 0 is rotated about the x axis find the area of the resulting surface or show that the area is infinite 5 Round Robin What method would you use to evaluate each of the following integrals Evaluate as many as possible in the time remaining R 2 a 0 sin2 z dz R 2 b x x 1 ex dx R 4 c 0 sec x tan x 1 sec2 x dx R0 d sin 2 cos d R e dx 4x x2 R f sec2 t dt R4 g 0 16 x2 dx R h sin ln t dt R 2 i x 2 x 2 dx R j sec u du
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