A Volume Example

with approximations

In this example the volume of the solid of revolution formed by revolving the graph of y = x^2 + 2 about the x-axis over the x-interval [-1,2] is investigated.

>	with(plots):

Warning, the name changecoords has been redefined

The region being revolved about the x-axis is pictured below.

>	plot(x^2+2,x=-1..2,filled=true);

The surface of revolution is graphed below.

>	surface1:=implicitplot3d(y^2+z^2=(x^2+2)^2,x=-1..2,y=-6..6,z=-6..6,axes=boxed,orientation=[225,45]):

>	display(surface1);

The volume of the associated solid is computed below from a definite integral using the Fundamental Theorem of Calculus to find the exact volume.  A decimal approximation is also computed.

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

>

value(%);

>	Vol:=evalf(%);

The loop below constructs 12 cylinders approximating the surface of revolution.  The cylinders are formed by partitioning the interval [-1,2] into 12 equal sized subintervals and using the midpoint of each subinterval in computing the radius of each cylinder.

>	cylind:=Array(1..12):

>	for i from 1 to 12 do

>	cylind[i]:=implicitplot3d(y^2+z^2=(2+(-1+(3/12)*(i-0.5))^2)^2,x=-1.25+i*0.25..-1+i*0.25,y=-6..6,z=-6..6,axes=boxed,orientation=[225,45]):

>

end do:

>	display(seq(cylind[k],k=1..12));

The sum of the volumes within each of the 12 approximating cylinders is computed below.

>	sum(Pi*((-1+(3/12)*(n-0.5))^2+2)^2*(3/12),n=1..12);

>

Vol;

The loop below constructs 24 cylinders approximating the surface of revolution.  The cylinders are formed by partitioning the interval [-1,2] into 24 equal sized subintervals and using the midpoint of each subinterval in computing the radius of each cylinder.

>	cylind:=Array(1..24):

>	for i from 1 to 24 do

>	cylind[i]:=implicitplot3d(y^2+z^2=(2+(-1+(3/24)*(i-0.5))^2)^2,x=-1.125+i*0.125..-1+i*0.125,y=-6..6,z=-6..6,axes=boxed,orientation=[225,45]):

>

end do:

>	display(seq(cylind[k],k=1..24));

The sum of the volumes within each of the 24 approximating cylinders is computed below.

>	sum(Pi*((-1+(3/24)*(n-0.5))^2+2)^2*(3/24),n=1..24);

>

Vol;

The loop below constructs 48 cylinders approximating the surface of revolution.  The cylinders are formed by partitioning the interval [-1,2] into 48 equal sized subintervals and using the midpoint of each subinterval in computing the radius of each cylinder.

>	cylind:=Array(1..48):

>	for i from 1 to 48 do

>	cylind[i]:=implicitplot3d(y^2+z^2=(2+(-1+(3/48)*(i-0.5))^2)^2,x=-1.0625+i*0.0625..-1+i*0.0625,y=-6..6,z=-6..6,axes=boxed,orientation=[225,45]):

>

end do:

>	display(seq(cylind[k],k=1..48));

The sum of the volumes within each of the 48 approximating cylinders is computed below.

>	sum(Pi*((-1+(3/48)*(n-0.5))^2+2)^2*(3/48),n=1..48);

>

Vol;

The sums below compute the sums of the volumes within each of 100, 1,000, 10,000, and 100,000 approximating cylinders.

>	sum(Pi*((-1+(3/100)*(n-0.5))^2+2)^2*(3/100),n=1..100);

>	sum(Pi*((-1+(3/1000)*(n-0.5))^2+2)^2*(3/1000),n=1..1000);

>	sum(Pi*((-1+(3/10000)*(n-0.5))^2+2)^2*(3/10000),n=1..10000);

>	sum(Pi*((-1+(3/100000)*(n-0.5))^2+2)^2*(3/100000),n=1..100000);

Of course the volume of the solid of revolution is the limit of this sum as n approaches infinity.

In the graph below we see the 12 approximating cylinders graph displayed along with a wireframe rendition of the surface.

>	surface2:=implicitplot3d(y^2+z^2=(x^2+2)^2,x=-1..2,y=-6..6,z=-6..6,axes=boxed,orientation=[225,45],style=wireframe,thickness=2):

>	display(surface2,seq(cylind[k],k=1..12));

>