Chapter 5: Applications of Integration
Section 5.3: Volume by Slicing
Example 5.3.1
By the method of slicing, obtain the volume of a wedge cut from a cylinder of radius r.
In particular, let the axis of symmetry for the cylinder lie along the z-axis, the bottom face of the wedge in the plane z=0, and the slanted face of the wedge in the plane that passes through the origin and that makes an angle α with the horizontal.
Solution
Mathematical Solution
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 x; the vertical, h.
The hypotenuse of the yellow (right) triangle has length r; the lengths of the legs are y and x.
From the yellow triangle, x2=r2−y2.
From the red triangle,
tanα=hx⇒h=x tanα
The area of the red triangle is A, given by
use DocumentTools, plots, plottools in
module()
local p1,p2,p3,p4,p5,p6,p7,p8,p9,p10;
p1 := display(polygon([[0,.5,0],[sqrt(3)/2,.5,0],[sqrt(3)/2,.5,.5]],color=red),transparency=.5):
p2 := spacecurve([[0,0,0],[sqrt(3)/2,.5,0]],color=black,thickness=2):
p3 := textplot3d([.5,.2,0,typeset(r)]):
p4 := textplot3d([.35,.2,.15,typeset(x)]):
p5 := textplot3d([.95,.5,.2,typeset(h)]):
p6 := textplot3d([.13,.48,.05,typeset(alpha)]):
p7 := display(polygon([[0,0,0],[0,.5,0],[sqrt(3)/2,.5,0]],color=yellow),transparency=.7):
p8 := textplot3d([.07,.3,0,typeset(y)]):
p9 := spacecurve([[0,0,0],[0,.5,0]],color=black,thickness=3):
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);
print(p10);
end module;
end use:
Figure 5.3.1(a) One slice in the wedge
A
=12 x h
=12 xx tanα
=12 x2 tanα
=12 r2−y2 tanα
Hence, the volume of the wedge is given by
V=tanα2∫−rrr2−y2 ⅆy=23tanαr3
use plots, plottools in
local q,q1,q2,q3,q4,qq;
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:
q1 := spacecurve([cos(t),sin(t),0],t=-Pi/2..Pi/2,color=black):
q2 := spacecurve([0,t,0],t=-1..1,color=black,thickness=3):
q3 := spacecurve([sqrt(1-y^2),y,sqrt(1-y^2)/sqrt(3)],y=-1..1,color=black):
qq := display([q1,q2,q3]):
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]);
print(q4);
end module:
Figure 5.3.1(b) Animation of slices
Figure 5.3.1(b) animates the slices within the wedge.
Maple Solution
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
tanα2∫−rrr2−y2 ⅆy = 23tanαr3
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