Final+Review - Math 2B 1 Final Review Integrals(a The...

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Math 2BFinal Review06/03/161.Integrals
(c)Fundamental Theorem of CalculusSupposefis continuous on [a, b].FTC I.Zxaf(t)dtforaxb, theng0(x) =f(x) on (a, b) andg(x) is continuous on[a, b].
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Math 2BFinal Review06/03/16Here is a table of indefinite integrals we need to know:Zcf(x)dx=cZf(x)dxZ[f(x)±g(x)]dx=Zf(x)dx±Zg(x)dxZkdx=kx+CZxndx=xn+1n+ 1+C(n6=-1)Z1xdx= ln|x|+CZexdx=ex+CZbxdx=bxlnb+CZsinxdx=-cosx+CZcosxdx= sinx+CZsec2xdx= tanx+CZcsc2xdx=-cotx+CZsecxtanxdx= secx+CZcscxcotxdx=-cscx+CZ1x2+ 1dx= tan-1x+CZ11-x2dx= sin-1x+C(d)u-SubstitutionIfu=g(x),u-substitution is given byZf(g(x))g0(x)dx=Zf(u)dufor an indefinite integralandZbaf(g(x))g0(x)dx=Zg(b)g(a)f(u)dufor a definite integral.Remember to change back toxfor indefinite integrals and to change your endpoints but not changeback toxfor definite integrals.2
Math 2BFinal Review06/03/16(e)Area Between Curves & Volumesi. The area between the curvesy=f(x) andy=g(x) betweenx=aandx=bisA=Zba|f(x)-g(x)|dx.Note: If these functions are defined in terms ofx, the integrand is Upper function - Lowerfunction. If these functions are defined in terms ofy, the integrand is Rightmost function -Leftmost function. Sometimes there is more than one of each of these.ii. The volume of a solid of revolution is given byV=ZbaA(x)dx,whereA(x) is the area of across section. For convenience, these come up often:If the cross-section is a disk, we find the radius (in terms ofxsay) and useA(x) =πr(x)2.If the cross-section is a washer, we find the inner radiusrinand outer radiusrout(in termsofx) and useA(x) =πrout(x)2-πrin(x)2.Note: This is notthe same asπ(rout(x)-rin(x))2!(f)Average ValueThe average value off(x) on [a, b] isfavg=1b-aZbaf(x)dx(g)Integration by PartsLetu=f(x),v=g(x). Thendu=f0(x)dx,dv=g0(x)dx. The formula for integration byparts isZudv=uv-Zvdu.In general:(a)Ifexis a multiplier in the integrand, you want to setdv=exdxas it is easy to integrate.(b)If lnx, tan-1x, or some function you don’t know the integral of is a multiplier, you wantto setu= lnxor whatever as it is easy to differentiate and difficult to integrate.(c)

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