Sec 11.1 Curves defined by Parametric Equations
DEFINITIONS:
A parametric curve is determined by a pair of
parametric equations: x = f (t), y = g (t), where f and
g are continuous on an interval a ≤ t ≤ b.
The variable t is called a parameter.
The points P = (f(a), g(a)) and Q = (f(b), g(b)) are called
the initial and terminal points, respectively.
The direction in which the curve is traced for increasing
values of the parameter is called the orientation of the
curve.
Note: A curve can have different parametrizations.
DEFINITIONS:
If the endpoints of the curve are the same, the curve is
called closed.
If distinct values of t yield distinct points in the plane
(except possibly for t = a and t = b), we say the curve
is a simple curve.
A simple closed curve is a curve that is both (i) closed
and (ii) simple.
A curve: x = f (t), y = g (t), a ≤ t ≤ b is called smooth
if f ′ and g′ exist and are continuous on [a, b], and f ′(t)
and g′(t) are not simultaneously zero on (a, b).
A Cycloid:
A cycloid is the curve traced by a point P on the
circumference of a circle as the circle rolls along a
straight line without slipping. See
http://www.ies.co.jp/math/java/calc/cycloid/cycloid.html
If a circle has radius r and rolls along the x-axis, and if
one position of P is the origin, the parametric equations
for the cycloid are
x  r (t  sin t ), y  r (1  cos t ), t  .
Sec 11.2
Calculus with Parametric Curves
Theorem: Let f and g be continuously differentiable
with f ′(t) ≠ 0 on α < t < β. Then the parametric
equations: x = f (t), y = g (t)
define y as a differentiable function of x and
dy
dy
g (t )
dt


dx
dx
f (t )
dt
Second Derivatives:
Theorem:
If the equations x = f (t), y = g (t) define y as a
twice-differentiable function of x, then at any
point where dx/dt ≠ 0,
d  dy 
 
2
d y
d  dy  dt  dx 

 
2
dx
dx
dx  dx 
dt
Area under a curve y = F(x) from a to b:
Theorem:
If the curve is traced out once by the parametric
equations: x = f (t), y = g (t), α ≤ t ≤ β, then the area
can be calculated by
A

b
a
y dx 


g (t ) f (t ) dt or



g (t ) f (t ) dt .

Arc Length
Theorem: The arc length of a smooth curve C given
by x = f (t), y = g (t), α ≤ t ≤ β, which does not
intersect itself except possibly at the endpoint is given
by
L 





2
2
 dx   dy 
     dt
 dt   dt 
 f (t )2  g (t )2
dt.
Surface Area
Theorem: If the curve C given by: x = f (t), y = g (t),
α ≤ t ≤ β, is rotated about the x-axis, where f and g
have continuous first derivatives, and g(t) ≥ 0, then the
area of the resulting surface is given by
S 


2
2
 dx   dy 
2 y      dt
 dt   dt 