MapleArcSA.mws

Arc Length and Surface Area

an example

>    with(plots): 

Warning, the name changecoords has been redefined

>    curve1:=plot([t,t^2-2*t+2,t=1..3],thickness=3,labels=[x,y]):

>    axesOfRev:=plot([0,t,t=-1..6],thickness=3,color=blue,linestyle=DASH):

>    display(curve1,axesOfRev);

[Maple Plot]

>    f:=x^2-2*x+2;

f := x^2-2*x+2

>    fprime:=diff(f,x);

fprime := 2*x-2

Below we compute the arc length for the curve above in red.

>    Int(sqrt(1+fprime^2),x=1..3);

Int((5+4*x^2-8*x)^(1/2),x = 1 .. 3)

>    value(%);

17^(1/2)-1/4*ln(-4+17^(1/2))

>    evalf(%);

4.646783762

Below we picture and compute the surface area of the surface formed by revolving the arc in red shown above about the blue dashed line (y-axis).

>    implicitplot3d(x^2+z^2=(1+sqrt(y-1))^2,x=-3..3,y=1..5,z=-3..3,axes=boxed,numpoints=2500);

[Maple Plot]

>    Int(2*Pi*(1+sqrt(y-1))*sqrt(1+1/(4*y-4)),y=1..5);

Int(2*Pi*(1+(y-1)^(1/2))*(1+1/(4*y-4))^(1/2),y = 1 .. 5)

>    value(%);

-1/6*Pi+29/6*Pi*17^(1/2)+1/4*Pi*ln(33+8*17^(1/2))

>    evalf(%);

65.37350668

>    Int(2*Pi*x*sqrt(1+(2*x-2)^2),x=1..3);

Int(2*Pi*x*(5+4*x^2-8*x)^(1/2),x = 1 .. 3)

>    value(%);

-1/8*Pi^(1/2)*(-68/3*Pi^(1/2)*17^(1/2)+4/3*Pi^(1/2))-1/4*Pi^(1/2)*(Pi^(1/2)+(-6*ln(2)-1)*Pi^(1/2)-8*Pi^(1/2)*17^(1/2)-2*Pi^(1/2)*ln(1/2+1/8*17^(1/2)))

>    evalf(%);

65.37350668

The integrations below appear to be done numerically.

>    int(2*Pi*(1+sqrt(y-1))*sqrt(1+1/(4*y-4)),y=1.0..5.0);

65.37350666

>    int(2*Pi*x*sqrt(1+(2*x-2)^2),x=1.0..3.0);

65.37350666

>