Return To Contents
Go To Problems & Solutions
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]
[1.8] [1.9] 
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 tinterval clearly doesn't consist of just a single point.
Return To Top Of Page Go To Problems & Solutions
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 xAxis, 

Fig.
2.2
Revolution
About Line y = k,


Fig.
2.3
Revolution
About yAxis,


Fig.
2.4
Revolution
About Line 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

[2.2] [2.3] [2.4] [2.5] [2.6] 
Find the area of the surface generated by revolving the parametric curve x = (t 1)^{2}, y = (8/3)t^{3/2} 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
= t^{2} 2t + 2,
discriminant = (2)^{2}
4(1)(2) = 4 < 0, so h(t) = t^{2} 2t + 2 is never 0,
h(0) = 2 > 0, thus h(t) > 0 for all t,
consequently r(t) = t^{2} 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 tlimits,
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.
Problems & Solutions 
1. Find the arc length of the parametric curve x = t^{2}, y = (2/3)(2t + 1)^{3/2} from t = 0 to t = 3.
let u = sin 2t; then du = 2 cos 2t dt, then:
^{{1}} Section 13.1.1 Example 3.4.
4. Determine the area of the surface generated by
revolving the curve represented parametrically by x
= t, y
= t^{2} + 1 from t
=
0 to t
= 3 about the yaxis.
5. Find the area of the surface of revolution
obtained by revolving the parametric curve x = 2t 4, y = t^{2} 3t from t = 3 to
t = 4 about
the line x = 4.
Return To Top Of Page Return To Contents