parametric equations and polar coordinatesfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · a...
TRANSCRIPT
![Page 1: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/1.jpg)
PARAMETRIC EQUATIONS
AND POLAR COORDINATES
10
![Page 2: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/2.jpg)
PARAMETRIC EQUATIONS & POLAR COORDINATES
We have seen how to represent curves by
parametric equations.
Now, we apply the methods of calculus to
these parametric curves.
![Page 3: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/3.jpg)
10.2
Calculus with
Parametric Curves
In this section, we will:
Use parametric curves to obtain formulas for
tangents, areas, arc lengths, and surface areas.
PARAMETRIC EQUATIONS & POLAR COORDINATES
![Page 4: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/4.jpg)
TANGENTS
In Section 10.1, we saw that some curves
defined by parametric equations x = f(t) and
y = g(t) can also be expressed—by eliminating
the parameter—in the form y = F(x).
See Exercise 67 for general conditions
under which this is possible.
![Page 5: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/5.jpg)
If we substitute x = f(t) and y = g(t)
in the equation y = F(x), we get:
g(t) = F(f(t))
So, if g, F, and f are differentiable,
the Chain Rule gives:
g’(t) = F’(f(t))f’(t) = F’(x)f’(t)
TANGENTS
![Page 6: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/6.jpg)
If f’(t) ≠ 0, we can solve
for F’(x):
'( )'( )
'( )
g tF x
f t
TANGENTS Equation 1
![Page 7: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/7.jpg)
The slope of the tangent to the curve y = F(x)
at (x, F(x)) is F’(x).
Thus, Equation 1 enables us to find
tangents to parametric curves without
having to eliminate the parameter.
TANGENTS
![Page 8: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/8.jpg)
Using Leibniz notation, we can rewrite
Equation 1 in an easily remembered form:
if 0
dy
dy dxdtdxdx dt
dt
TANGENTS Equation 2
![Page 9: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/9.jpg)
If we think of a parametric curve as
being traced out by a moving particle,
then
dy/dt and dx/dt are the vertical and horizontal
velocities of the particle.
Formula 2 says that the slope of the tangent
is the ratio of these velocities.
TANGENTS
![Page 10: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/10.jpg)
From Equation 2, we can see that
the curve has:
A horizontal tangent when dy/dt = 0
(provided dx/dt ≠ 0).
A vertical tangent when dx/dt = 0
(provided dy/dt ≠ 0).
TANGENTS
![Page 11: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/11.jpg)
TANGENTS
This information is useful for
sketching parametric curves.
![Page 12: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/12.jpg)
As we know from Chapter 4, it is also useful
to consider d2y/dx2.
This can be found by replacing y by dy/dx
in Equation 2:
2
2
d dy
d y d dy dt dx
dxdx dx dx
dt
TANGENTS
![Page 13: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/13.jpg)
A curve C is defined by the parametric
equations x = t2, y = t3 – 3t.
a. Show that C has two tangents at the point (3, 0)
and find their equations.
b. Find the points on C where the tangent is
horizontal or vertical.
c. Determine where the curve is concave upward
or downward.
d. Sketch the curve.
TANGENTS Example 1
![Page 14: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/14.jpg)
Notice that y = t3 – 3t = t(t2 – 3) = 0
when t = 0 or t = ± .
Thus, the point (3, 0) on C arises from
two values of the parameter: t = and t = –
This indicates that C crosses itself at (3, 0).
3
3
TANGENTS Example 1 a
3
![Page 15: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/15.jpg)
Since
the slope of the tangent when t = ±
is:
dy/dx = ± 6/(2 ) = ±
2/ 3 3 3 1
/ 2 2
dy dy dt tt
dx dx dt t t
TANGENTS Example 1 a
3
33
![Page 16: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/16.jpg)
So, the equations of the tangents
at (3, 0) are:
3 3
and
3 3
y x
y x
TANGENTS Example 1 a
![Page 17: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/17.jpg)
C has a horizontal tangent when dy/dx = 0,
that is, when dy/dt = 0 and dx/dt ≠ 0.
Since dy/dt = 3t2 - 3, this happens when t2 = 1,
that is, t = ±1.
The corresponding points on C are (1, -2) and (1, 2).
TANGENTS Example 1 b
![Page 18: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/18.jpg)
C has a vertical tangent when
dx/dt = 2t = 0, that is, t = 0.
Note that dy/dt ≠ 0 there.
The corresponding point on C is (0, 0).
TANGENTS Example 1 b
![Page 19: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/19.jpg)
To determine concavity, we calculate
the second derivative:
The curve is concave upward when t > 0.
It is concave downward when t < 0.
22 2
2 3
3 11
3 12
2 4
d dytd y dt dx t
dxdx t t
dt
TANGENTS Example 1 c
![Page 20: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/20.jpg)
Using the information from (b) and (c),
we sketch C.
TANGENTS Example 1 d
![Page 21: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/21.jpg)
a. Find the tangent to the cycloid
x = r(θ – sin θ), y = r(1 – cos θ )
at the point where θ = π/3.
See Example 7 in Section 10.1
b. At what points is the tangent horizontal?
When is it vertical?
TANGENTS Example 2
![Page 22: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/22.jpg)
The slope of the tangent line is:
/ sin sin
/ 1 cos 1 cos
dy dy d r
dx dx d r
TANGENTS Example 2 a
![Page 23: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/23.jpg)
When θ = π/3, we have
and
3sin
3 3 3 2
1 cos3 2
x r r
ry r
1
2
sin / 3 3 / 23
1 cos / 3 1
dy
dx
TANGENTS Example 2 a
![Page 24: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/24.jpg)
Hence, the slope of the tangent is .
Its equation is:
3
33
2 3 2
or
3 23
r r ry x
x y r
TANGENTS Example 2 a
![Page 25: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/25.jpg)
The tangent is sketched here.
TANGENTS Example 2 a
![Page 26: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/26.jpg)
The tangent is horizontal when
dy/dx = 0.
This occurs when sin θ = 0 and 1 – cos θ ≠ 0,
that is, θ = (2n – 1)π, n an integer.
The corresponding point on the cycloid is
((2n – 1)πr, 2r).
Example 2 b TANGENTS
![Page 27: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/27.jpg)
When θ = 2nπ, both dx/dθ and dy/dθ are 0.
It appears from the graph that there
are vertical tangents at those points.
TANGENTS Example 2 b
![Page 28: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/28.jpg)
We can verify this by using l’Hospital’s Rule
as follows:
2 2
2
sinlim lim
1 cos
coslim
sin
n n
n
dy
dx
TANGENTS Example 2 b
![Page 29: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/29.jpg)
A similar computation shows
that dy/dx → -∞ as θ → 2nπ – .
So, indeed, there are vertical tangents
when θ = 2nπ, that is, when x = 2nπr.
TANGENTS Example 2 b
![Page 30: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/30.jpg)
AREAS
We know that the area under a curve y = F(x)
from a to b is
where F(x) ≥ 0.
( )b
aA F x dx
![Page 31: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/31.jpg)
Suppose the curve is traced out once by
the parametric equations x = f(t) and y = g(t),
α ≤ t ≤ β.
Then, we can calculate an area formula by using
the Substitution Rule for Definite Integrals.
( ) '( )
or ( ) '( )
A y dx g t f t dt
g t f t dt
AREAS
![Page 32: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/32.jpg)
Find the area under one arch
of the cycloid
x = r(θ – sin θ) y = r(1 – cos θ)
AREAS Example 3
![Page 33: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/33.jpg)
One arch of the cycloid is given by 0 ≤ θ ≤ 2π.
Using the Substitution Rule with
y = r(1 – cos θ) and dx = r(1 – cos θ) dθ,
we have the following result.
AREAS Example 3
![Page 34: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/34.jpg)
AREAS
2
0
2
0
2 22
0
22 2
0
212
20
23 1 32 2 2
2 4 20
= 1 cos 1 cos
1 cos
1 2cos cos
1 2cos 1 cos 2
2sin sin 2 .2 3
r
A y dx
r r d
r d
r d
r d
r r r
Example 3
![Page 35: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/35.jpg)
AREAS
The result of Example 3 says that the area
under one arch of the cycloid is three times
the area of the rolling circle that generates
the cycloid (Example 7 in Section 10.1).
Galileo guessed this result.
However, it was first proved by the French
mathematician Roberval and the Italian
mathematician Torricelli.
Example 3
![Page 36: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/36.jpg)
ARC LENGTH
We already know how to find the length L
of a curve C given in the form
y = F(x), a ≤ x ≤ b
![Page 37: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/37.jpg)
ARC LENGTH
Formula 3 in Section 8.1 says that,
if F’ is continuous, then
2
1b
a
dyL dx
dx
Equation 3
![Page 38: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/38.jpg)
Suppose that C can also be described by
the parametric equations x = f(t) and y = g(t),
α ≤ t ≤ β, where dx/dt = f’(t) > 0.
This means that C is traversed once, from left to right,
as t increases from α to β and f(α) = a and f(β) = b.
ARC LENGTH
![Page 39: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/39.jpg)
Putting Formula 2 into Formula 3 and using
the Substitution Rule, we obtain:
2
2
1
/1
/
b
a
dyL dx
dx
dy dt dxdt
dx dt dt
ARC LENGTH
![Page 40: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/40.jpg)
Since dx/dt > 0, we have:
2 2dx dy
L dtdt dt
ARC LENGTH Formula 4
![Page 41: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/41.jpg)
Even if C can’t be expressed in the form
y = f(x), Formula 4 is still valid.
However, we obtain it by polygonal
approximations.
ARC LENGTH
![Page 42: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/42.jpg)
We divide the parameter interval [α, β]
into n subintervals of equal width Δt.
If t0, t1, t2, . . . , tn are the endpoints of these
subintervals, then xi = f(ti) and yi = g(ti) are
the coordinates of points Pi(xi, yi) that lie on C.
ARC LENGTH
![Page 43: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/43.jpg)
So, the polygon with vertices P0, P1 , ..., Pn
approximates C.
ARC LENGTH
![Page 44: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/44.jpg)
As in Section 8.1, we define the length L of C
to be the limit of the lengths of these
approximating polygons as n → ∞:
1
1
limn
i in
i
L P P
ARC LENGTH
![Page 45: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/45.jpg)
The Mean Value Theorem, when applied to f
on the interval [ti–1, ti], gives a number ti*
in (ti–1, ti) such that:
f(ti) – f(ti–1) = f’(ti*)(ti – ti–1)
If we let Δxi = xi – xi–1 and Δyi = yi – yi–1,
the equation becomes:
Δxi = f’(ti*) Δt
ARC LENGTH
![Page 46: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/46.jpg)
Similarly, when applied to g, the Mean
Value Theorem gives a number ti**
in (ti–1, ti) such that:
Δyi = g’(ti**) Δt
ARC LENGTH
![Page 47: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/47.jpg)
Therefore,
2 2
1
2 2* **
2 2* **
' '
' '
i i i i
i i
i i
P P x y
f t t g t t
f t g t t
ARC LENGTH
![Page 48: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/48.jpg)
Thus,
2 2
* **
1
lim ' 'n
i in
i
L f t g t t
ARC LENGTH Equation 5
![Page 49: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/49.jpg)
The sum in Equation 5 resembles a Riemann
sum for the function .
However, is not exactly a Riemann sum
because ti* ≠ ti
** in general.
2 2
' 'f t g t
ARC LENGTH
![Page 50: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/50.jpg)
Nevertheless, if f’ and g’ are continuous,
it can be shown that the limit in Equation 5
is the same as if ti* and ti
** were equal,
namely,
2 2
' 'L f t g t dt
ARC LENGTH
![Page 51: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/51.jpg)
Thus, using Leibniz notation, we
have the following result—which has
the same form as Formula 4.
ARC LENGTH
![Page 52: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/52.jpg)
Let a curve C is described by
the parametric equations x = f(t), y = g(t),
α ≤ t ≤ β, where:
f’ and g’ are continuous on [α, β].
C is traversed exactly once as t increases from α to β.
Then, the length of C is:
2 2dx dy
L dtdt dt
THEOREM Theorem 6
![Page 53: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/53.jpg)
Notice that the formula in Theorem 6
is consistent with these general formulas
from Section 8.1:
L = ∫ ds and (ds)2 = (dx)2 + (dy)2
ARC LENGTH
![Page 54: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/54.jpg)
Suppose we use the representation
of the unit circle given in Example 2 in
Section 10.1:
x = cos t y = sin t 0 ≤ t ≤ 2π
Then,
dx/dt = –sin t and dy/dt = cos t
ARC LENGTH Example 4
![Page 55: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/55.jpg)
So, as expected, Theorem 6 gives:
2 22 2
2 2
0 0
2
0
sin cos
2
2
dx dydt t tdt
dt dt
dt
ARC LENGTH Example 4
![Page 56: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/56.jpg)
On the other hand, suppose we use
the representation given in Example 3 in
Section 10.1:
x = sin 2t y = cos 2t 0 ≤ t ≤ 2π
Then,
dx/dt = 2cos 2t and dy/dt = –2sin 2t
ARC LENGTH Example 4
![Page 57: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/57.jpg)
Then, the integral in Theorem 6 gives:
2 22 2
2 2
0 0
2
0
4cos 2 4sin 2
2
4
dx dydt t t dt
dt dt
dt
ARC LENGTH Example 4
![Page 58: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/58.jpg)
Notice that the integral gives twice
the arc length of the circle.
This is because, as t increases from 0 to 2π,
the point (sin 2t, cos 2t) traverses the circle twice.
ARC LENGTH Example 4
![Page 59: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/59.jpg)
In general, when finding the length of a curve
C from a parametric representation, we have
to be careful to ensure that C is traversed only
once as t increases from α to β.
ARC LENGTH Example 4
![Page 60: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/60.jpg)
Find the length of one arch
of the cycloid
x = r(θ – sin θ) y = r(1 – cos θ)
From Example 3, we see that one arch is described
by the parameter interval 0 ≤ θ ≤ 2π.
ARC LENGTH Example 5
![Page 61: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/61.jpg)
We have:
1 cos
and
sin
dxr
d
dyr
d
ARC LENGTH Example 5
![Page 62: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/62.jpg)
Thus,
2 22
0
2 22 2 2
0
22 2 2
0
2
0
1 cos sin
r 1 2cos cos sin
2 1 cos
dx dyL d
d d
r r d
d
r d
ARC LENGTH Example 5
![Page 63: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/63.jpg)
To evaluate this integral, we
use the identity
with θ = 2x.
This gives 1 – cos θ = 2sin2(θ/2).
12
2sin 1 cos2x x
ARC LENGTH Example 5
![Page 64: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/64.jpg)
Since 0 ≤ θ ≤ 2π, we have 0 ≤ θ/2 ≤ π,
and so sin(θ/2) ≥ 0.
Therefore,
22 1 cos 4sin / 2
2 sin / 2
2sin / 2
ARC LENGTH Example 5
![Page 65: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/65.jpg)
Hence,
2
0
2
0
2 sin / 2
2 2cos / 2
2 2 2
8
L r d
r
r
r
ARC LENGTH Example 5
![Page 66: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/66.jpg)
The result of Example 5 says that the length of
one arch of a cycloid is eight times the radius
of the generating circle.
This was first proved in 1658 by Sir Christopher Wren.
ARC LENGTH
![Page 67: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/67.jpg)
SURFACE AREA
In the same way as for arc length,
we can adapt Formula 5 in Section 8.2
to obtain a formula for surface area.
![Page 68: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/68.jpg)
Let the curve given by the parametric
equations x = f(t), y = g(t), α ≤ t ≤ β,
be rotated about the x-axis, where:
f’, g’ are continuous.
g(t) ≥ 0.
Then, the area of the resulting surface
is given by: 2 2
2dx dy
S y dtdt dt
SURFACE AREA Formula 7
![Page 69: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/69.jpg)
The general symbolic formulas
S = ∫ 2πy ds and S = ∫ 2πx ds (Formulas 7
and 8 in Section 8.2) are still valid.
However, for parametric curves,
we use: 2 2dx dy
ds dtdt dt
SURFACE AREA
![Page 70: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/70.jpg)
Show that the surface area of a sphere
of radius r is 4πr2.
The sphere is obtained by rotating the semicircle
x = r cos t y = r sin t 0 ≤ t ≤ π
about the x-axis.
SURFACE AREA Example 6
![Page 71: PARAMETRIC EQUATIONS AND POLAR COORDINATESfac.ksu.edu.sa/sites/default/files/chap10_sec2.pdf · A curve C is defined by the parametric equations x = t2, y = t3 ± 3t. a. Show that](https://reader033.vdocuments.us/reader033/viewer/2022041420/5e1e394201000a1c9952220d/html5/thumbnails/71.jpg)
So, from Formula 7, we get:
*
2 2
0
2 2 2
0
0
2
0
2 2
0
2 sin sin cos
2 sin sin cos
2 sin
2 sin
2 cos 4
S r t r t r t dt
r t r t t dt
r t r dt
r t dt
r t r
SURFACE AREA Example 6