Solids of Revolution

Let's begin by plotting functions of one-variable:

Plot[(x^2) - (x^4), {x, 0,1}]

Plot[Sin[], {x, 0.02,1}]

Plot[x*Sin[(1/x)], {x, -1,1}, PlotStyle->{RGBColor[0,1,0], Thickness[.005]}]

In the above plot, what happens if you change the Thickness parameter (which should be a number between 0 and 1)?
What happens if you change the RGB mix?  (RGB is an abbreviation for "Red, Green, Blue")

ParametricPlot[{(t^2), Sqrt[t]+1}, {t,  0,2}]

Now, for several more exciting examples:

ParametricPlot[{Sin[3 t+ Pi/4], Sin[t]}, {t, 0, 2 Pi}, PlotStyle ->{Red,Thickness[0.01]}]

ParametricPlot[{Cos[t]-(((Sin[t])^2)/(Sqrt[2])), Cos[t] Sin[t]}, {t, 0, 2 Pi}, PlotStyle ->{Blue,Thickness[0.01]}]

ParametricPlot[{(∫_0^t)Sin[(u^2)]\[DifferentialD]u, (∫_0^t)Cos[(u^2)]\[DifferentialD]u}, {t, -5, 5}, PlotStyle ->{Red}]

Implicitly defined functions

For implicitly defined functions, we use the ContourPlot command. Notice the use of the double equal sign!

ContourPlot[(x^2)-(y^2)==(x^3),{x,-2,2},{y,-2,2}, Axes->True]

ContourPlot[(x^3)+(y^3)==6x y,{x,-5,5},{y,-5,5}, Axes->True]

This implicitly defined graph is called the Folium of Descartes.

Solids of Revolution

Recall our first example in which we plotted the curve z = (x^2)- (x^4)  and considered the region bounded by this curve and the x-axis.  Let's revolve this region about the z-axis

RevolutionPlot3D[(x^2)- (x^4),{x, 0,1}, AxesLabel->{x,y,z}]

The default is to rotate about  the z-axis  (which, in fact, is our y-axis). 
Next, we will rotate about the x-axis

RevolutionPlot3D[(x^2)- (x^4),{x, 0,1},RevolutionAxis-> {1,0,0}, AxesLabel->{x,y,z}]

Rotate about line z = x

RevolutionPlot3D[(x^2)- (x^4),{x, 0,1},RevolutionAxis->{1,0,1}, AxesLabel->{x,y,z}]

RevolutionPlot3D[2 Sin[x],{x, 0,4 Pi},RevolutionAxis-> {1,0,0}, AxesLabel->{x,y,z}]

RevolutionPlot3D[4 Sin[x],{x, 0,4 Pi}, AxesLabel->{x,y,z}]

RevolutionPlot3D[Sin[x],{x, 0,Pi}, AxesLabel->{x,y,z}]

RevolutionPlot3D[Sqrt[1-x^2],{x, -1,1},RevolutionAxis->{1,0,0}, AxesLabel->{x,y,z}]

RevolutionPlot3D[x,{x, 0,5}, AxesLabel->{x,y,z}, BoxRatios->{1,1,1}]

RevolutionPlot3D[1/(x^2),{x, 0,9}, AxesLabel->{x,y,z}, BoxRatios->{1,1,1}]

RevolutionPlot3D[Sin[x],{x, 0,5 Pi},RevolutionAxis-> {0,0,1}, AxesLabel->{x,y,z}]

Solids of revolution determined by parametrically defined curves.

ParametricPlot[{t+1,Sin[t-5]}, {t,  0,Pi}, AxesLabel->{x,y}]

RevolutionPlot3D[{t+1,Sin[t-5]},{t, 0, Pi},AxesLabel->{x,y,z}]

ParametricPlot[{4+Cos[t],Sin[t]},{t,0,2Pi}, AxesOrigin-> {0,0}]

RevolutionPlot3D[{4+Cos[t],Sin[t]},{t,0,2Pi}]

ParametricPlot[{Sin[t]+Sin[9t]/5,Cos[t]+Cos[9t]/5},{t,Pi/2,Pi}]

Let us observe what happens when we rotate this self-intersecting curve about the z-axis.

RevolutionPlot3D[{Sin[t]+(1/5) Sin[9 t],Cos[t]+(1/5) Cos[9 t]},{t,Pi/2,π},AxesLabel->{x,y,z}]

Computing Volume of a Solid of Revolution

Consider our very first example:  y =  (x^2) - (x^4)  rotated about the x-axis, 0≤x≤1.

Using disks, the volume is (∫_0^1)π ((((x^2) - (x^4)))^2)\[DifferentialD]x.  Mathematica does the computation for us:

(∫_0^1)π ((((x^2) - (x^4)))^2)\[DifferentialD]x

((8 π)/315)

To convert this to an approximate decimal:

N[%]

If we rotate this region about the y-axis, we should use shells:

(∫_0^1)2 π x((x^2) - (x^4))\[DifferentialD]x

N[%]

Piecewise Continuous Functions

Suppose that we wish to plot a piecewise continouous function such as  f(x) = (x^2) if x <1 and f(x)=2x -1 if x>=1.  Mathematica uses the following command:

f[x_]:=Piecewise[{{(x^2),x<1},{x/2+1/2,x≥1}}]

Show[Plot[f[x], {x,-3,5}]]

Let us rotate the piecewises continuous function about the x-axis:

RevolutionPlot3D[f[x], {x,-2,2}, RevolutionAxis-> {1,0,0}]