Fall09-Pfinal-sol - MAT126 Fall 2009 Practice Final The actual Final exam will consist of twelve problems 1 Problem 1 2 Evaluate 1 Evaluate sin2

Fall09-Pfinal-sol - MAT126 Fall 2009 Practice Final The...

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MAT126 Fall 2009Practice FinalThe actual Final exam will consist of twelve problems1
Problem 11. Evaluateintegraltextπ/2π/3sin3(x) cos2(x)dx2. Evaluateintegraltextsin2(x)dxSolution:1.First, recognize that the integrand consists of sin(x) raised to an odd power,multiplied by cos(x) raised to an even power. Such a function is a prime candidate forthe substitutionu=cos(x) (so that one of the sin(x) can be used for the change ofvariable, and the rest may be expressed in terms of cos(x) via the identity sin2(x)=1cos2(x)).Therefore we substituteu=cos(x), wherebydu=sin(x)dx, and we can write(noting thatu=cos(π/3)=1/2 whenx=π/3, andu=cos(π/2)=0 whenx=π/2)integraldisplayx=π/2x=π/3sin3(x) cos2(x)dx=integraldisplayx=π/2x=π/3(sin2(x) cos2(x))(sin(x)dx)=integraldisplayx=π/2x=π/3(1+cos2(x))(cos2(x))(sin(x)dx)=integraldisplayu=0u=1/2(u21)(u2)du=integraldisplayu=0u=1/2(u4u2)du=u55u33vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingleu=0u=1/2=parenleftBigg(0)55(0)33parenrightBiggparenleftBigg(1/2)55(1/2)33parenrightBigg=255+2332.There are two ways to approach this problem. The first is to use the identitycos(2x)=cos2(x)sin2(x)=12 sin2(x)to getsin2(x)=12(1cos(2x))2
This implies thatintegraldisplaysin2(x)dx=integraldisplay12(1cos(2x))dx=integraldisplay12dxintegraldisplay12cos(2x)dx=12x12integraldisplaycos(2x)dx=12x12sin(2x)2+C=12x14sin(2x)+CThe second approach is to use an integration by parts, settingu=sinxanddv=sinxdx; this means thatdu=cosxdxandv=cosx. Therefore, integration by partsgivesintegraldisplaysin2(x)dx=(sinx)(cosx)integraldisplay(cosx)(cosxdx)=sinxcosx+integraldisplaycos2(x)dxNow we plug in the identity cos2(x)=1sin2(x) to getintegraldisplaysin2(x)dx=sinxcosx+integraldisplay(1sin2(x))dx=sinxcosx+integraldisplaydxintegraldisplaysin2(x)dxAddingintegraltextsin2(x)dxto both sides of the equation gives2integraldisplaysin2(x)dx=sin(x) cos(x)+integraldisplaydx=sin(x) cos(x)+x+Cso thatintegraldisplaysin2(x)dx=12x12sin(x) cos(x)+CNotice that both approaches give the same answer, because of the identity sin(2x)=2 sin(x) cos(x).3
Problem 21. Estimate the integral8integraldisplay7dxln(x)using three rectangles and(a) right endpoints(b) left endpoints(c) Are your answers in 1a and 1b over- or under-estimates of the actual inte-gral?2. Do the same for the under-integral functionf(t)=et3Solution:1.The problem asks for three rectangles, so we divide the interval [7,8] into threeequal subintervals:bracketleftBigg7,713bracketrightBiggbracketleftBigg713,723bracketrightBiggbracketleftBigg723,8bracketrightBiggEach subinterval has length13.

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