December 2017 page 2 of 2 16 6 a sketch the curves x

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December 2017Page 2 of 2[16]6.a.Sketch the curvesx=y2andx=y3, and find the area enclosed.b.Find the volume of the solid obtained by rotating the region boundedby the curvex=y2+ 1 and the linex= 3about the linex= 3.c.Show thatRa0f(x)dx=Ra0f(a-x)dxfor any functionf(x) continuous on [0, a].[6]7.Find the limit of the sequence{an}asn→ ∞or prove that it does not exist:(a)an=enn!(b)an= (-1)nsin (n)n[12]8.Determine whether the series is divergent or convergent, and if convergent,whether absolutely or conditionally :(a)Xn=1(-1)nn+ 1n2n+ 2(b)Xn=1(-1)nln (n)n(c)Xn=0n+ 1-n[6]9.Find(a)the radius of convergence, and(b)the interval convergence of theseriesn=0(3x-2)n5n.[8]10.(a)Derive the Maclaurin series off(x) =11 +x2.(HINT: start with the series for11-zwherez=-x2).(b)Use integrability of the power series derived in (a) to find thepower series for arctan(x).[5]Bonus question.Use the formula given above in6 cto evaluateπ/2R0sinxsinx+ cosxdx.

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