Chapter 5: Applications of Integration
Section 5.3: Volume by Slicing
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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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Solution
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Mathematical Solution
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The red (right) triangle in Figure 5.3.1(a) is one slice in the wedge cut from the cylinder. The horizontal leg has length ; the vertical, .
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The hypotenuse of the yellow (right) triangle has length ; the lengths of the legs are and .
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From the yellow triangle, .
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The area of the red triangle is , given by
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use DocumentTools, plots, plottools in
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local p1,p2,p3,p4,p5,p6,p7,p8,p9,p10;
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p1 := display(polygon([[0,.5,0],[sqrt(3)/2,.5,0],[sqrt(3)/2,.5,.5]],color=red),transparency=.5):
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p2 := spacecurve([[0,0,0],[sqrt(3)/2,.5,0]],color=black,thickness=2):
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p3 := textplot3d([.5,.2,0,typeset(r)]):
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p4 := textplot3d([.35,.2,.15,typeset(x)]):
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p5 := textplot3d([.95,.5,.2,typeset(h)]):
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p6 := textplot3d([.13,.48,.05,typeset(alpha)]):
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p7 := display(polygon([[0,0,0],[0,.5,0],[sqrt(3)/2,.5,0]],color=yellow),transparency=.7):
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p8 := textplot3d([.07,.3,0,typeset(y)]):
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p9 := spacecurve([[0,0,0],[0,.5,0]],color=black,thickness=3):
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p10 := display([p1,p2,p3,p4,p5,p6,p7,p8,p9],axes=frame,orientation=[-105,75],labels=[x,y,z],tickmarks=[[0],0,0],view=[0..1,0..1,0..0.5],scaling=constrained);
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Figure 5.3.1(a) One slice in the wedge
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Hence, the volume of the wedge is given by
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use plots, plottools in
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local q,q1,q2,q3,q4,qq;
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q := proc(y)
display(polygon([[0,y,0],[sqrt(1-y^2),y,0],[sqrt(1-y^2),y,sqrt(1-y^2)/sqrt(3)]],color=red)):
end proc:
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q1 := spacecurve([cos(t),sin(t),0],t=-Pi/2..Pi/2,color=black):
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q2 := spacecurve([0,t,0],t=-1..1,color=black,thickness=3):
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q3 := spacecurve([sqrt(1-y^2),y,sqrt(1-y^2)/sqrt(3)],y=-1..1,color=black):
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qq := display([q1,q2,q3]):
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q4 := animate(q,[y],y=-1..1,axes=box, frames=51, background=qq, scaling=constrained, labels=[x,y,z],view=0..1,orientation=[-140,65],paraminfo=false,tickmarks=[0,[0],0]);
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Figure 5.3.1(b) Animation of slices
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Figure 5.3.1(b) animates the slices within the wedge.
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Maple Solution
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Expression palette: Definite-integral template
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Context Panel: Evaluate and Display Inline
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=
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