MapleSA.mws

Surface Area

of a surface in space

This worksheet investigates the area of the surface defined by f(x,y) = 9-x^2-y^2

over the region of the xy-coordinate plane defined by x between -2.5 and 2.5 and y between -0.5 and 2.5.

Plotting the surface

>    with(plots):surface:=plot3d(9-y^2-x^2,x=-2.5..2.5,y=-0.5..2.5,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    display(surface);

[Maple Plot]

Using one tangent plane to approximate the surface

>    Tnplane16:=plot3d(10-2*y,x=-2.5..2.5,y=-0.5..2.5,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    display(surface,Tnplane16);

[Maple Plot]

Using four tangent planes to approximate the surface

>    Tnplane20:=plot3d(109/8-(5/2)*x-(7/2)*y,x=0..2.5,y=1..2.5,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    Tnplane19:=plot3d(109/8+(5/2)*x-(7/2)*y,x=-2.5..0,y=1..2.5,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    Tnplane18:=plot3d(87/8-(5/2)*x-(1/2)*y,x=0..2.5,y=-0.5..1,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    Tnplane17:=plot3d(87/8+(5/2)*x-(1/2)*y,x=-2.5..0,y=-0.5..1,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    display(surface,Tnplane17,Tnplane18,Tnplane19,Tnplane20);

[Maple Plot]

Using fifteen tangent planes to approximate the surface and showing the partition in the xy-coordinate plane

>    B1:=spacecurve([2.5,-0.5+t,-4,t=0..3],color=blue):

>    B2:=spacecurve([1.5,-0.5+t,-4,t=0..3],color=blue):

>    B3:=spacecurve([0.5,-0.5+t,-4,t=0..3],color=blue):

>    B4:=spacecurve([-0.5,-0.5+t,-4,t=0..3],color=blue):

>    B5:=spacecurve([-1.5,-0.5+t,-4,t=0..3],color=blue):

>    B6:=spacecurve([-2.5,-0.5+t,-4,t=0..3],color=blue):

>    B7:=spacecurve([-2.5+t,-0.5,-4,t=0..5],color=blue):

>    B8:=spacecurve([-2.5+t,0.5,-4,t=0..5],color=blue):

>    B9:=spacecurve([-2.5+t,1.5,-4,t=0..5],color=blue):

>    B10:=spacecurve([-2.5+t,2.5,-4,t=0..5],color=blue):

>    Tnplane1:=plot3d(11-2*x-2*y,x=0.5..1.5,y=0.5..1.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane2:=plot3d(17-4*x-4*y,x=1.5..2.5,y=1.5..2.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane3:=plot3d(14-4*x-2*y,x=1.5..2.5,y=0.5..1.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane4:=plot3d(14-2*x-4*y,x=0.5..1.5,y=1.5..2.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane5:=plot3d(13-4*y,x=-0.5..0.5,y=1.5..2.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane6:=plot3d(10-2*y,x=-0.5..0.5,y=0.5..1.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane7:=plot3d(11+2*x-2*y,x=-1.5..-0.5,y=0.5..1.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane8:=plot3d(14+2*x-4*y,x=-1.5..-0.5,y=1.5..2.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane9:=plot3d(14+4*x-2*y,x=-2.5..-1.5,y=0.5..1.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane10:=plot3d(17+4*x-4*y,x=-2.5..-1.5,y=1.5..2.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane11:=plot3d(13+4*x,x=-2.5..-1.5,y=-0.5..0.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane12:=plot3d(10+2*x,x=-1.5..-0.5,y=-0.5..0.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane13:=plot3d(9,x=-0.5..0.5,y=-0.5..0.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane14:=plot3d(10-2*x,x=0.5..1.5,y=-0.5..0.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    Tnplane15:=plot3d(13-4*x,x=1.5..2.5,y=-0.5..0.5,view=-4..10,axes=boxed,orientation=[100,60]):

>    surface:=plot3d(9-y^2-x^2,x=-2.5..2.5,y=-0.5..2.5,view=-4..11.2,axes=boxed,orientation=[100,60]):

>    display(B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,Tnplane1,Tnplane2,Tnplane3,Tnplane4,Tnplane5,Tnplane6,Tnplane7,Tnplane8,Tnplane9,Tnplane10,Tnplane11,Tnplane12,Tnplane13,Tnplane14,Tnplane15,surface);

[Maple Plot]

Computing the area of the surface using an iterated double integral

>    int(int(sqrt(1+4*x^2+4*y^2),y=-0.5..2.5),x=-2.5..2.5);

56.91846271

Approximating the area of the surface from 15 approximating pieces of tangent planes (see picture above)

>    Yvals:=-1:

>    Area:=0:

>    for i from 1 to 3 do
  Yvals:=Yvals+1;
  Xvals:=-3;
  for j from 1 to 5 do
    Xvals:=Xvals+1;
    Area:=Area+sqrt(4*Xvals^2+4*Yvals^2+1);
  end do:
end do:

>    Area;

3*17^(1/2)+3*5^(1/2)+7+4*21^(1/2)+2*33^(1/2)

>    evalf(Area,8);

55.896949

Approximating the area of the surface from 60 approximating pieces of tangent planes

>    Yvals:=-0.75:

>    Area:=0:

>    for i from 1 to 6 do
  Yvals:=Yvals+0.5;
  Xvals:=-2.75;
  for j from 1 to 10 do
    Xvals:=Xvals+0.5;
    Area:=Area+(1/4)*sqrt(4*Xvals^2+4*Yvals^2+1);
  end do:
end do:

>    Area;

56.67582891

Approximating the area of the surface from 240 approximating pieces of tangent planes

>    Yvals:=-0.625:

>    Area:=0:

>    for i from 1 to 12 do
  Yvals:=Yvals+0.25;
  Xvals:=-2.625;
  for j from 1 to 20 do
    Xvals:=Xvals+0.25;
    Area:=Area+(1/16)*sqrt(4*Xvals^2+4*Yvals^2+1);
  end do:
end do:

>    Area;

56.85786513

Approximating the area of the surface from 960 approximating pieces of tangent planes

>    Yvals:=-0.5625:

>    Area:=0:

>    for i from 1 to 24 do
  Yvals:=Yvals+0.125;
  Xvals:=-2.5625;
  for j from 1 to 40 do
    Xvals:=Xvals+0.125;
    Area:=Area+(1/64)*sqrt(4*Xvals^2+4*Yvals^2+1);
  end do:
end do:

>    Area;

56.90332182

>