Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 13: Plane Curves – Section 13.1.3: Arc Length And Area Of Surface Of Revolution Of Parametric
Curves


13.1.3
Arc Length And Area Of Surface Of Revolution Of
Parametric Curves

 

 

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1. Arc Length Of Parametric Curves

 

The arc length of the graph of a continuously differentiable function was discussed in Section 12.6. Refer to Fig. 1.1. The arc
length s of the graph of the continuously differentiable function y = F(x) over [a, b] is:

 

 

[1.1]

 

 

 

 

[1.2]

 

 

 

 

[1.3]

 

Fig. 1.1

 

Arc Length And Differential Of Arc Length.

 

Now we're going to determine a formula for the arc length of a smooth parametric curve x = f(t), y = g(t), using the
parametric functions f and g, so that we don't have to first find the corresponding Cartesian function y = F(x) or equation G(x,
y) = 0. For the curve to be smooth with certainty, f '(t) and g'(t) must be continuous and never be simultaneously 0. Utilizing
the differential triangle equation as the starting point and manipulating it we have:

 

 

 

 

 

 

 

 

 

[1.4]

 

 

 

 

 

[1.5]

 

This is an easy way to remember the equation for arc length of parametric curves. However it can't be taken as a “proof ” of
the equation, because although Eq. [1.3] is valid for ordinary functions y = f(x), we don't know yet if it's also valid for
parametric curves. For the proof we rely on a sum that behaves like Riemann sums, and we employ chords as we did in
Section 12.6 Part 2.

 

 

Fig. 1.2

 

Finding Arc Length Of Parametric Curve.

 

 

{1.1} Section 12.6 Part 3.

 

Fig. 1.3

 

Differential Triangle.

 

 

The arc length s of the parametric curve x = f(t), y = g(t) from t = a to t = b is:

   

 

 

 

 

 

  [1.6]

 

 

 

 


  [1.7]

 

 

 

 

 

  [1.8]

 

 

 

 

  [1.9]

 

Example 1.1

 

 

Fig. 1.4

 

Curve For Example 1.1.

 

Solution

EOS

 

The absolute value must be handled appropriately. Otherwise we would get:

 

 

which of course is wrong, since this curve for this t-interval clearly doesn't consist of just a single point.

 

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2. Area Of Surface Of Revolution Of Parametric Curves

 

In Section 12.7 we investigated the area of surface of revolution for ordinary functions y = f(x). Refer to Figs. 2.1 thru to 2.4. In
general the area S of the surface of revolution for y = f(x) from x = a to x = b about a line parallel to an axis is:

 

 

[2.1]

 

Fig. 2.1

 

Revolution About x-Axis,
r(x) = | f(x)|.

 

Fig. 2.2

 

Revolution About Line y = k,
r(x) = | f(x) – k|.

 

Fig. 2.3

 

Revolution About y-Axis,
r(x) = |x|.

 

Fig. 2.4

 

Revolution About Line x = k,
r(x) = |x – k|.

 

Distinguish between small s for arc length and big S for area. In Fig. 2.1 the graph is revolved about the x-axis and the radius
of revolution is
r(x) = | f(x)|. In Fig. 2.2 the graph is revolved about the line y = k (parallel to the x-axis) and the radius of
revolution is
r(x) = | f(x) – k|. In Fig. 2.3 the graph is revolved about the y-axis and the radius of revolution is r(x) = |x|. In
Fig. 2.4 the graph is revolved about the line x = k (parallel to the y-axis) and the radius of revolution is r(x) = |x – k|.

 

Now we establish equations for area of surface of revolution of a parametric curve x = f(t), y = g(t) from t = a to t = b, using
the parametric functions f and g, so that we don't have to first find the corresponding Cartesian function y = F(x) or equation
G(x, y) = 0. The process is similar to that in Part 1. The differential of arc length ds is expressed in terms of the parameter t.
Thus the radius of revolution r must be expressed in terms or be a function of the parameter t too.

 

 

Let S be the area of the surface of revolution for the parametric curve x = f(t), y = g(t) from t = a to t = b about a
line parallel to an axis. Then:

 

 

 

 

 

 

 

 

[2.2]

 

[2.3]

 

[2.4]

 

 

 

 

 

[2.5]

 

 

 

[2.6]

 

Example 2.1


Find the area of the surface generated by revolving the parametric curve x = (t – 1)2, y = (8/3)t3/2 from t = 1 to t = 3 about
the line x = –1.

 

Solution
The radius of revolution at x corresponding to the parameter value t is r(t) = |x – (–1)| = |(t – 1)2 + 1| = |t2 – 2t + 2|,
discriminant = (–2)2 – 4(1)(2) = – 4 < 0, so h(t) = t2 – 2t + 2 is never 0,
h(0) = 2 > 0, thus h(t) > 0 for all t,
consequently r(t) = t2 – 2t + 2.

 

Now:

 


EOS

 

We don't have to draw a picture if not asked to. We just recall the appropriate picture from among Figs. 2.1 thru to 2.4
mentally, determine the radius of revolution, determine the differential of arc length, and integrate the differential of area
between appropriate t-limits, knowing that (differential of area) = (circumference of revolution) x (differential of arc length).

Keep in mind that all the quantities involved must be in terms of the parameter t.

 

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Problems & Solutions

 

1. Find the arc length of the parametric curve x = t2, y = (2/3)(2t + 1)3/2 from t = 0 to t = 3.

 

Solution

 

 

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Solution

 

 

let u = sin 2t; then du = 2 cos 2t dt, then:

 

 

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{1} Section 13.1.1 Example 3.4.

 

Solution

 

 

 

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4. Determine the area of the surface generated by revolving the curve represented parametrically by x = t, y = t2 + 1 from t =
    0 to t = 3 about the y-axis.

 

Solution

 

 

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5. Find the area of the surface of revolution obtained by revolving the parametric curve x = 2t – 4, y = t2 – 3t from t = 3 to
    t = 4 about the line x = 4.

 

Solution

 

 

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