Visualizing the Surface of Revolution

Note that this is just to help us visualize the process! In practice the sketches are easier to draw by hand, and should be drawn by hand with a single washer shown to determine the inner and outer radius.

You should not worry about the Mathematica code used to generate these plots, unless you find it interesting.

Revolution about Horizontal Line

What is below could be modified to revolve about a vertical line, but for the moment I will focus on a horizontal line. This is just meant to help us visualize what we are doing, since in practice we will be drawing sketches by hand to accomplish this task.

Let’s define the functions, and shade the region we want to rotate about the line y=β:

a=0;
b=1;
yT[x_]=(x^2)+1;
yB[x_]=x;
β=-1
Plot[{yT[x],yB[x],β},{x,a,b},AxesLabel->{"x","y"},PlotRange->{All,All},Filling->{1->{2}},BaseStyle->24,PlotStyle->{Black,Black,Red},FillingStyle->Blue]

So we are going to rotate the shaded region about the red line, creating a volume of revolution. The image above is in (ℝ^2) and we will want to include it in an image in (ℝ^3), so we will have to figure out how to do that in Mathematica.

The easiest way to include the original region being rotated in the sketch in (ℝ^3) is to use the ImplicitRegion and DiscretizingRegion commands in Mathematica. Again, this is only necessary to get Mathematica to produce the image, it is not necessary when solving the problem by hand. I do not expect you to know these commands.

region=ImplicitRegion[yB[x]≤y≤yT[x]&&z==0,{x,y,z}];
DiscretizeRegion[region,{{a,b},{-7,5},{-1,1}}];
region3D=RegionPlot3D[%,PlotStyle->Blue]

If you rotate this image you can see it is now in (ℝ^3).

Draw Some Washers

To draw some of the washers, we need to set up a partition of the interval x∈[a,b] using n partiitions.

(*define the partiaition--needed for the image below, but not needed when problem is solved by hand.*)
Clear[xp]
n=4;
Δx=(b-a)/n;
xp[0]=a;
Table[xp[i]=xp[i-1]+Δx,{i,1,n+1}];

(*define the innner and outer radii--use the diagram to figure these radii out. We do this when we solve the problem by hand.*)
router[x_]=-β+yT[x];
rinner[x_]=-β+yB[x];

(*this creates the images of the washers--by hand, we would just draw one washer to create the formula.*)
washers=Table[
RegionPlot3D[(z^2)+(((y-β))^2)≤((router[xp[i]])^2)&&(z^2)+(((y-β))^2)≥((rinner[xp[i]])^2),{x,xp[i-1],xp[i]},{y,-4,2},{z,-4,4},PlotStyle->Directive[Yellow,Opacity[0.45]],Mesh->None]
,{i,1,n}];

(*This creates the line we are rotating about, using a parametric plot*)
line=ParametricPlot3D[{t,β,0},{t,a,b},PlotStyle->{Thickness[0.01],Red}];

Show[washers,line,region3D,PlotRange->{All,All,All},BaseStyle->18,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0}]

Rotate the image to view it from different angles (you can do this by clicking on the image with your mouse and dragging).

Notice the washers are centered around the line y=β (the red line in the diagram).

Notice how the region being rotated was used to create the washers with right hand endpoints.

The volume of the washers above is an approximation to the volume we seek.

If we decrease the width of the washers, we get a better approximation to the volume we seek.

(*define the partiaition--needed for the image below, but not needed when problem is solved by hand.*)
Clear[xp]
n=35;
Δx=(b-a)/n;
xp[0]=a;
Table[xp[i]=xp[i-1]+Δx,{i,1,n+1}];

(*define the innner and outer radii--use the diagram to figure these radii out. We do this when we solve the problem by hand.*)
router[x_]=-β+yT[x];
rinner[x_]=-β+yB[x];

(*this creates the images of the washers--by hand, we would just draw one washer to create the formula.*)
washers=Table[
RegionPlot3D[(z^2)+(((y-β))^2)≤((router[xp[i]])^2)&&(z^2)+(((y-β))^2)≥((rinner[xp[i]])^2),{x,xp[i-1],xp[i]},{y,-4,2},{z,-4,4},PlotStyle->Directive[Yellow,Opacity[0.25]],Mesh->None,BoundaryStyle->None]
,{i,1,n}];

(*This creates the line we are rotating about, using a parametric plot*)
line=ParametricPlot3D[{t,β,0},{t,a,b},PlotStyle->{Thickness[0.01],Red}];

Show[washers,line,region3D,PlotRange->{All,All,All},BaseStyle->18,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0}]

Rotate the image to view it from different angles (you can do this by clicking on the image with your mouse and dragging).

Finding the Volume

The sketch we would draw by hand would look something like this (when we do it by hand we can label it better!):

(*define the partiaition--needed for the image below, but not needed when problem is solved by hand.*)
Clear[xp]
n=50;
Δx=(b-a)/n;
xp[0]=a;
Table[xp[i]=xp[i-1]+Δx,{i,1,n+1}];

(*define the innner and outer radii--use the diagram to figure these radii out. We do this when we solve the problem by hand.*)
router[x_]=-β+yT[x];
rinner[x_]=-β+yB[x];

(*this creates the images of the washers--by hand, we would just draw one washer to create the formula.*)
i=25
washers=
RegionPlot3D[(z^2)+(((y-β))^2)≤((router[xp[i]])^2)&&(z^2)+(((y-β))^2)≥((rinner[xp[i]])^2),{x,xp[i-1],xp[i]},{y,-4,2},{z,-4,4},PlotStyle->Directive[Yellow,Opacity[0.25]],Mesh->None,BoundaryStyle->None];

(*This creates the line we are rotating about, using a parametric plot*)
line=ParametricPlot3D[{t,β,0},{t,a,b},PlotStyle->{Thickness[0.01],Red}];

Show[washers,line,region3D,PlotRange->{All,All,All},BaseStyle->18,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0}]

And here is how we set up the formula for the volume.

(Subscript[y, T]) = (x^2)+1
(Subscript[y, B])=x
a=0 b=1 creates the region in x y plane (these will give the limits in the integral at the end).

Outer radius = 1 +(Subscript[y, T])=1+(x^2)+1=2+(x^2)
Inner radius = 1 +(Subscript[y, B])=1+x

Area of washer = π (((Outer radius))^2)-π (((Inner radius))^2)=π((((2+(x^2)))^2)-(((1+x))^2))=π (3-2 x+3 (x^2)+(x^4))

Volume of washer = π (3-2 x+3 (x^2)+(x^4))dx 

Volume we seek  = (∫_0^1)π (3-2 x+3 (x^2)+(x^4))dx

Integrate[π (3-2 x+3 (x^2)+(x^4)),{x,0,1}]