# mat1322e-2018-1-notes-03-filled-in-updated.pdf -...

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StuDocu is not sponsored or endorsed by any college or universityMat1322e 2018 1 notes 03 filled in updatedCalculus II (University of Ottawa)StuDocu is not sponsored or endorsed by any college or universityMat1322e 2018 1 notes 03 filled in updatedCalculus II (University of Ottawa)Downloaded by lina she ([email protected])lOMoARcPSD|3552958
3. Volumes(Jan. 15)We’ve now seen how definite integrals give us a way to compute areas of “weird”shapes bounded by functions.Let’s generalize these concepts to help us find volumes of 3-dimensional shapes.VOLUMESConsider a solid object that lies betweenx=aandx=b. What is its volume?solidApproximatevolumeusingslices..pt#EE....¥÷⇐€EE¥E¥EE¥⇐¥¥¥¥¥÷E¥÷I¥÷¥¥÷E¥E¥⇐Ev=nkjma§lAk*D×=fabtyxldx1Downloaded by lina she ([email protected])
Example 3.1.Find the volume of the solid obtained by rotating about thex-axis the regionunder the curvey=xfrom0to1.Example 3.2.Derive the formula for the volume of a cone of heightHand radiusR.2^^^^#y=rx=)MakeT<o•G!'G.<o.¥÷¥*i.><o.*i.>G.<.¥E÷*}YYs**.\thitgknxessK¥5VvVvDX••atypicalsliceinilhsubintervaliscircularwithapproximateradiusRkitkfxixandthicknessDX=¥=thapprox.volumeofonesliceisIRZDX=I§xf*TD×approxvdumeofentiresolidisV=⇐§AkDax=§jKtF*TDx.:'V=SottFYdx=fkxdx=t¥]j=If⇒-F(E)=E2^#,^nRnsimilarDs}Hi**iiz*}HRyb±I=¥,<><>slicetheconeintoncircularslicesOfthicknessDy=Hn€Atypicalsliceinithsubintervalhasapproximateradius=Rlyi)Gbyusingsimilar