# Theorem 6 method of washers if the region between y f

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Theorem 6(Method of Washers).If the region betweeny=f(x)andy=g(x)fromx=atox=bis rotated around the liney=c, then, by
18MATH 2B GUIDE TO LECTURESdetermining (from the graph) the outer radiusroutand the inner radiusrin(as functions ofx), the volume of the solid of revolution isV=Example: Find the volume of the solid obtained by rotating aboutthex-axis the region under the curvey=xfrom 0 to 1.
MATH 2B GUIDE TO LECTURES19Section 6.5Guiding questions:How is the definite integral connected to a function’s averagevalue?
Theorem 7.The average value offon the interval[a, b]isfave=Example: Find the average value off(x) = 1+x2on the interval[-1,2].
-13-1You try: Assume an object moves in a straight line with velocityv(t) = sintmeters/second.Find the average velocity on the timeinterval [0, π/2].
20MATH 2B GUIDE TO LECTURESSection 7.1Guiding questions:How can integrate a product of functions?How can we integrate a function when we only know a deriva-tive?
Theorem 8(Integration by parts).If the integrand can be writtenf(x)g0(x), then, settingu=f(x)anddv=g0(x)dx, we haveZudv=andZbaudv=uv]ba-Zbavdu.Idea: The goal is to make the new integral simpler than the original.We also must selectdvso that we know its antiderivative.Example: FindRxsinxdx.
MATH 2B GUIDE TO LECTURES21General Strategy: For selectingu: LIATE: in order of preference:L:I:A:T:E:Another idea: Sometimes, we know aderivativebut not an integral.Example: FindRlnx dx.Solution:We could use LIATE, but using the technique just mentioned: weknow the derivative of lnxbut not the integral.u= lnx,du=1xdxdv=dx,v=xZlnx dx=xlnx-Z1xx dx=xlnx-Zdx=xlnx-x+C.You try: FindRtan-1x dx.
22MATH 2B GUIDE TO LECTURES0.1.Section 7.2.Guiding questions: