Calculus II PLTLFall 2014Worksheet 6These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Scribe) Use the substitution x = 3 sin θ to evaluateRdx/(x2√9 − x2).2) (Pairs) (a) Approximate the integralR60(1 − x2) dx usi ng the following methods:(i) Trapezoidal Rule with n = 4(ii) Simpson’s Rule with n = 4(b) What i s the exact value of I =R60(1 − x2) dx?(c) How does the exact value I compare to the approximations in (a)? Does therelationship b etween S4and I surprise you? Loo k at how the er ror bound for S4iscalculated a nd explain the relationship. For what types of functions wil l this relationshiphold?(d) Why is the relationship between T4and I not as strong? Can you think of anytypes of functions for which the erro r bound for Tnwill always be zero?(e) What i s the smallest value of n such that |Tn−I| < 0.001?3) (Round Robin) Cons ider the i ntegralR0−5xx2+4x−5dx.(a) Explain what an improper integral is in your own words, and dis cuss the ap-proaches you can take when evaluatin g 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 =xx2+4x−5onthe interval [−5, 0]?(d) If possible, determine the value of the integral.4) (Scribe) If the infinite curve y = e−x, x ≥ 0, is rotated about the x-axis, find the areaof the resulting surface, or show that the area i s infinite.5) (Round Robin) What method wo uld you use to eva luate each o f the following integrals ?Evaluate as many as possible in the time r emaining.(a)Rπ/20sin2z dz(b)R(x2+ x + 1)exdx(c)Rπ/40(sec x tan x)/(1 + sec2x) dx(d)R0−π(sin θ)/(2 + cos θ) dθ(e)Rdx/√4x − x2(f)Rsec2t dt(g)R40√16 − x2dx(h)Rsin(ln t) dt(i)R(x2+ 2)/(x + 2) dx(j)Rsec u
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