Chapter 5: Applications of Integration
Section 5.3: Volume by Slicing
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Essentials
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The animation in Figure 5.3.1 shows a cutting plane intersecting a solid.
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Suppose the slice at exposes a region with area .
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Further, suppose is "thickened" by to a slab; the volume of this slab would be .
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The volume of the solid would be the sum of the volumes of the slabs, that is
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>
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G:=proc(s)
local p1,p2,p3,p4;
p1:=plots:-implicitplot3d((x/9)^2+(y/5)^2+(z/3)^2=1,x=-9..9,y=-5..5,z=-3..3,scaling=constrained,style=surface,axes=frame,tickmarks=[0,0,0],labels=[x,"",""],transparency=.5,color=red):
p2:=plots:-implicitplot3d(x=s,x=-9..9,y=-5..5,z=-3..3,style=surface,color=green):
plots:-display(p1,p2);
end proc:
plots:-animate(G,[x],x=-9..9,orientation=[50,70,0],frames=11,paraminfo=false);
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Figure 5.3.1 Solid segmented by slicing
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Examples
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Example 5.3.1
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By the method of slicing, obtain the volume of a wedge cut from a cylinder of radius . In particular, let the axis of symmetry for the cylinder lie along the -axis, the bottom face of the wedge in the plane , and the slanted face of the wedge in the plane that passes through the origin and that makes an angle with the horizontal.
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Example 5.3.2
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By the method of slicing, obtain the volume of the solid whose base is an equilateral triangle of side , and whose plane sections are squares. In particular, the equilateral triangle lies in the plane and has a vertex at the origin, and an altitude along the -axis. The square cross sections are parallel to the -plane.
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Example 5.3.3
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By the method of slicing, obtain the volume of the solid whose base in the -plane is the region bounded by the -axis, and the curves and , and whose cross sections parallel to the -plane are equilateral triangles.
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