# Marks consider the function f x e x ² x2 1³ a

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5.[7marks]Consider the functionf(x) =ex²x2+ 1³.(a) Determine the absolute minimum value and the absolute maximum value offon the interval[±2;0].(b) Using your answer to part(a), and the properties of de°nite integrals, show that54<0R°2ex²x2+ 1³dx²26.[8marks]Use the limit of an appropriate Riemann sum to determine the value of:6R2(x+ 3)2dx.4
7.[6marks]Find the area of the region bounded by the curvesy2=x+ 1andx=y+ 1.
8.[OMIT- Volumes not covered this term]LetRdenote the region bounded between the curvesy=x2andy=px.Find the volume of the solid obtained by rotatingRabout the linex= 4.-2246-2-112345
9.[5marks]Use a substitution to evaluate1xpx+ 3dx.10.[7marks]Use integration by parts to evaluate2R1tan°1px±1dx.6R6
11.[12marks]Determine(x±2) (x2+ 2x+ 3)dxby the method of partial fraction expansion.
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12.[6marks]Show that the two-variable functionu= sin (x±at) + ln (x+at)satis°es thewave equationutt=a2uxx(whereais a constant).13.[8marks]Find all critical points and determine the relative extrema (if any) off(x; y) =x3±3xy+ 3y2±9y.Use the Second Derivative Test to justify your conclusions.8
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