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...Triple integrals for volumes of some classic shapes in the following pages i give worked out examples where are used to nd boxes cylinders spheres and cones all these aren t needed but just want show you how could use them methods cylindrical spherical coordinates also illustrated hope this helps better understand set up a integral remember that volume solid region e is given by zzz dv arectangular box can be described inequalities x b c y d p z q so comes length times width height as expected bz dz qdzdydx acircular cylinder equation outer edge circular radius if we consider inside such with h then considering other words between planes already have bounds on let s innermost now need xy plane do few dierent ways cartesian thesolidcanbedescribedbytheinequalities dzdydx dx note skipped steps integration would see last geomet rically represents area exactly half circle or trig substitution perfect very easily together r az rdzdrd rdr either way get formula asphere sphere regions will thr...