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Parametric Equations
Curve C: y  f (x) (C fails the Vertical Line Test.)
The x-coordinate and the y-coordinate are functions of time, i.e.,
x  f (t )
y  g (t ) , where t is called a parameter.
Curve C (Parametric curve) : ( x, y )  ( f (t ), g (t ))
at b
Initial point ( f (a ), g (a )) , terminal point ( f (b), g (b))
Examples : Sketch and identify the curve defined by the parametric equations
(1)
x  t 2  2t
y  t 1
(2)
x  1  3t
y  2  t2
Examples: Eliminate the parameter to find a Cartesian equation of the curve. Describe the
curve.
(1) x  cos t
(2) x  sin 2t
y  sin t
0  t  2
y  cos 2t
0  t  2
(3) x  et  1 y  e 2t
(4) x  ln t
(5) x  3 sec 
y t
t 1
y  4 tan 
Example: Find parametric equation for the circle with center (h,k) and radius r.
Example: Sketch the curve with parameter equations x  sin t
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y  sin 2 t .
Spring 2010
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Calculus with Parametric Curves
y  F ( x),
x  f (t ),
y  g (t )
We get g (t )  F ( f (t )) . Use the Chain Rule, we have
g (t )  F ( f (t )) f (t )
'
'
'
g ' (t )
 If f (t )  0 , then F ( x)  '
f (t )
'
'
The slope of the tangent to the curve y  F (x ) at ( x, F ( x)) is F ' ( x) .
dy
dy dt
Leibniz Notation:

dx dx
dt
if
dx
0
dt
♦ When
dy
dx
 0 , the curve has a horizontal tangent. (
 0)
dt
dt
♦ When
dx
dy
 0 , the curve has a vertical tangent. (
 0)
dt
dt
d  dy 
 
d y d  dy  dt  dx 


Moreover,
 
dx
dx 2 dx  dx 
dt
2
Example 1: C: x  t 2 ,
(a)
(b)
(c)
(d)
y  t 3  3t
Show that C has two tangents at the point (3,0) and find their equations.
Find the point on C where the tangent is horizontal or vertical.
Determine whether the curve is concave upward or downward.
Sketch the curve.
Example 2: Find the equation of the tangent to the curve at the point corresponding to
x  cos   sin 2 , y  sin   cos 2 .
  0.
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b

a

Areas: A   ydx   g (t ) f ' (t )dt

2
2
2
 dy 
 dx   dy 
1    dx        dt
 dx 
 dt 
  dt 
b
Arc length: L  
a
2

2
2
 dy 
 dx   dy 
Surface area: S   2y 1    dx   2y      dt (About the x-axis)
 dx 
 dt   dt 
a

b
Example: Use the parametric equation of an ellipse,
x  a cos  , y  b sin  , 0    2 , to find the area that it encloses.
Example: Cycloid : x  r   sin   y  r 1  cos  
(a) Find the area under one arch of the cycloid, i.e., 0    2 .
(b) Find the length of one arch of the cycloid.
Example: Find the surface area of a sphere of radius r. (Answer. 4r 2 )
(Hint: Rotating the semicircle x  r cos t
y  r sin t
0  t   about the x-axis.)
Example: Find the exact area of the surface obtained by rotating the given curve about
the x-axis. x  3t  t 3 , y  3t 2 , 0  t  1.
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