1120_5 - Week 5 Outline More volumes by slicing Procedures...

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Week 5OutlineMore volumes by slicing–Procedures of Calculating the volume of a Solid–Some examples that are not solids of revolutionCompute the arc length of a curve–the definition of arc length–the formula of arc lengthProcedures of Calculating the Volume of a Solid(1) Sketch the solid and a typical cross-section.(2) Find a formula forA(x), the area of a typical cross-section.(3) Find the lower and upper limits of integration.(4) IntegrateA(x).Cavalieri’s PrincipleSolids with equal altitudes and identical cross-sectional areas at each height have thesame volume.Examples1. A square of side lengthslies in a plane perpendicular to a lineL. One vertex ofthe square lies onL. As this square moves a distancehalongL, the square turns onerevolution aboutLto generate a corkscrew-like column with square cross-sections.(a) Find the volume of the column.(b) What will the volume be if the square turns twice instead of once?2. The solid lies between planes perpendicular to thex-axis atx=-1 andx

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