Laboratory 19
Arc Length and Surfaces of Revolution

Introduction

    Given a function y = f(x), a≤x≤b, we see how to find the length of the curve from (a, f(a)) to (b, f(b)). If this curve is revolved about an axis, it generates a surface of revolution,  and we see how to find the resulting surface area.

Example 19.1, Arc Length

    Mathematica is particularly useful for finding the length of a curve determined by a function y = f(x). It can be shown that if y = f(x) is a differentiable function, then the length of the arc of the curve determined by this function from  x = a to x = b is
                                                      ∫_a^b (1 + (y ')^2)^(1/2)  x = ∫_a^b (1 + (f ' (x))^2)^(1/2)  x.
Other than some special cases, such integrals are impossible to calculate without using numerical methods, so that Mathematica is especially useful. Before we calculate arc lengths, let us evaluate some integrals of the above type. We start with the function y = (2/3) x^(3/2), from x = 0 to x = 1. The integral to be calculated is then ∫_0^1 (1 + x)^(1/2)  x, since y ' = x^(1/2).

Integrate[Sqrt[1 + x], x]

(2/3 + (2 x)/3) (1 + x)^(1/2)

     We see that the Integrate command will find the anti-derivative when it exists. The actual value of the arc length between (0, 0) and (1, 2/3) is then:.

Integrate[Sqrt[1 + x], {x, 0, 1}]

-2/3 + (4 2^(1/2))/3

    Suppose we want to find the length of the arc of the sine curve y = sin (x) from (0, 0) to (π, 0).

Clear[f] f[x_] := Sin[x]

Plot[f[x], {x , 0,   Pi}, PlotStyle->RGBColor[1, 0, 0]] ;

[Graphics:HTMLFiles/index_26.gif]

Integrate[Sqrt[1 + (f '[x])^2], x]

2^(1/2) EllipticE[x, 1/2]

Mathematica is not able to give a simple answer as no anti-derivative of (1 + cos^2(x))^(1/2) in terms of elementary functions exists. However, Mathematica can give an approximate numerical value for the integral as follows:

NIntegrate[Sqrt[1 + (f '[x])^2], { x, 0, Pi}]

3.8202

Experiment 19.1

    For each of the following curves y = f(x), determine whether or not the integrand in the arc length expression has an anti-derivative in terms of elementary functions. In each case determine a numerical value for the arc length between the x -values indicated.    

    1.  f(x) = x^3/3 + 1/(4x),  a = 1, b = 2.

    2.  f(x) = (1 - x^2)^(1/2), a = 0, b = 1.
    
    3. f(x) = arctan (x), a = 0, b = 1.  
    
    4. f(x) = 3x^2 - 1, a = 0, b = 1.
    
    5. f(x) = ln(x), a = 1, b = e.

Example 19.2, Surfaces of Revolution  

     If the function y = f(x) is rotated about a line, the surface area of the solid of revolution which is formed is given by the formula:

     S = 2π∫_a^br(x) (1 + (f ' (x))^2)^(1/2)  x,

where r(x) is the distance between the graph of f(x) and the axis of revolution (so we require r(x) ≥0). When the rotation is about the x -axis, r(x) = f(x), but when the rotation is about the y -axis, r(x) = x. As an example, let us rotate y = sin (x) about the x -axis, between the lines x = 0, and x = π. In this case r(x) = sin (x). We can use the module Rev introduced in the lab on volumes of revolution, having calling parameters f, a and b, to obtain an idea what the surface looks like.

Rev[f_, a_, b_] := Module[{t}, ParametricPlot3D[{x, f[x] Cos[t], f[x] Sin[t]}, <br />         {x, a, b}, {t, 0, 2 Pi}] ;]

Clear[f, x] f[x_] := Sin[x]

Rev[f, 0, Pi]

[Graphics:HTMLFiles/index_65.gif]

The surface area is then:

Integrate[2Pi f[x] Sqrt[1 + (f '[x])^2], {x, 0, Pi}]

2 π (2^(1/2) + ArcSinh[1])

or rather:

NIntegrate[2Pi f[x] Sqrt[1 + (f '[x])^2], {x, 0, Pi}]

14.4236

Experiment 19.2

    Find the surface area of the solid of revolution obtained when the part of the curve y = f(x) between the lines x = a and x = b is rotated. Give a 3D parametric plot of the surfaces in each case. (N0TE - You will need the ParametricPlot3D command to complete this experiment).    

           1.  f(x) = (8x)/(1 + x^2), a = 0, b = 10. (Rotate about the x -axis and use NIntegrate).

    2.  f(x) = ln(x), a = 1, b = e.  (Rotate about the y -axis and recall that Mathematica uses Log[x] for the natural log function).
    
    3. f(x) = arctan(x), a = 0, b = 1.
    
    Now do the rotation about the line x = 3, for exercise 2.

Example 19.3, Surface Area and Volume of a Torus  

    If we rotate a circle about the y -axis, we obtain the doughnut shape known as a torus. We can illustrate this by rotating the top and bottom halves of the circle (x - 2)^2 + y^2 = 1. To graph the region to be revolved we use:

Clear[f, g] f[x_] := Sqrt[1 - (x - 2)^2] g[x_] := -Sqrt[1 - (x - 2)^2]

Plot[{f[x], g[x]}, {x, 1, 3}, AspectRatio->Automatic, AxesLabel-> {"x", "y"}, PlotRange {{0, 3}, All}] ;

[Graphics:HTMLFiles/index_86.gif]

Now rotate the circle about the y-axis:

A = ParametricPlot3D[{x Cos[t], x Sin[t], f[x]}, {x, 1, 3}, {t, 0, 2 Pi}, DisplayFunctionɳ ... 2 Pi}, DisplayFunctionIdentity] ; Show[{A, B}, DisplayFunction$DisplayFunction] ;

[Graphics:HTMLFiles/index_89.gif]

    This solid has volume V = 4π ∫_1^3x f(x)  x, which we can use Mathematica to evaluate:

Integrate[4 Pi x f[x], {x, 1, 3}]

4 π^2

    The surface area is given by Σ = 4π ∫_1^3x (1 +[f ' (x)]^2)^(1/2)  x, which we evaluate as follows:

Integrate[4 Pi x Sqrt[1 + f '[x]^2], {x, 1, 3}]

8 π^2

Experiment  19.3

1. Using different values for a and b (a<b), find the volume V and surface area Σ obtained by rotating the circle (x - b)^2 + y^2 = a^2  about the y -axis, and hence conjecture a formula for the volume and surface area of a torus obtained by rotating a circle of radius a about a line distance b from the center of the circle. (Optional exercise: Prove that your formula is correct using calculus).

2. Use Mathematica to determine the formula for the volume of the torus in Experiment 1 for general values of a and b (Mathematica will perform the symbolic integration necessary for the volume, but may cause some difficulty with the surface area.).

Credits

These laboratories were created by Raouf Boules, Geoff Goodson, Ohoe Kim and Mike O'Leary for use in the Calculus courses of Towson University. Commercial use is prohibited without permission of the authors. Non-commercial use is permitted, provided this credit section is retained.

Created by Mathematica  (August 20, 2004)