10
10.1
Parametric Equations and Polar Coordinates.
Curves Defined by Parametric Equations.
1. parametric equations (as equations of a third variable)
2. parametric equations with initial and terminal point
3. curve (set of points) versus parametric curve (curve with an orientation)
4. cartesian equation (equation of the form f (x1 , x2 , . . . xn ) = 0. i.e. without the
extra parameter t. Example: x2 + y 2 − 1 = 0 or also x2 + y 2 = 1)
5. cycloid (the curve traced out by a fixed point on the circumference of a rotating
circle along a straight line) with parametric equations x = r(θ − sin θ) and
y = r(1 − cos θ), with θ ∈ R
6. sketch parametric equations by combining x and y in order to eliminate the
third variable (t or θ). For the not so obvious ways to do so, we’ll use Section
10.2.
7. useful formulas in solving some trig integrals: sin2 x =
1 + cos(2x)
2
1 − cos(2x)
and cos2 x =
2
8. length of an arc = rθ, where θ is in radians
10.2
Calculus with Parametric Curves
1. tangents to parametric curves (slopes)
2. second derivative
Z
3. area A =
d2 y
b
dx
Z
a
dy
dx
=
dy
dt
dx
dt
β
ydx =
when a ≤ x ≤ b
=
dy
)
d( dx
dt
dx
dt
g(t)f 0 (t)dt, where x = f (t) and y = g(t) and α ≤ t ≤ β
α
Z βs
Z br
Z β r 2 2
dy 2
dy 2 dx
dx
dy
dt
dx =
+
dt
4. arc length L =
1+
1 + dx
dt =
dx
dt
dt
dt
a
α
α
dt
where x = f (t) and y = g(t) a nd α ≤ t ≤ β when a ≤ x ≤ b
r Z b
Z β
dx 2 dy 2
5. surface area of the curve y rotating with the x-axis S =
2πyds =
2πy
+
dt
dt
dt
a
α
where x = f (t) and y = g(t) and α ≤ t ≤ β when a ≤ x ≤ b
10.3
Polar Coordinates
1. cartesian coordinates (x and y values of a point in the x − y plane)
2. polar coordinates (polar system has a pole (the origin) and a polar axis (the
positive x-axis). And so the polar coordinates of P are r = the distance(O, P ),
and θ = the angle, in radians, that OP maxes with the polar axis)
3. cartesian coordinates in terms of polar coordinates: x = r cos θ, y = r sin θ
p
y
4. polar coordinates in terms of cartesian coordinates: r = x2 + y 2 , tan θ =
x
5. the graph of a polar equations r = f (θ) or F (r, θ) = 0 consists of the points
that have polar coordinates r, θ
6. when sketching the curve r = f (θ), first draw the graph of f (θ) and use it to
get the curve.
7. polar equation unchanged when θ is replaced by −θ, then curve is symmetric
with the polar axis
8. polar equation unchanged when r is replaced by −r or when θ is replaced by
θ + π, then curve is symmetric about the pole (i.e. the curve is the same if it is
rotated 1800 )
9. polar equation unchanged when θ is replaced by π − θ, then curve is symmetric
about the vertical line θ = π2
dy
10. the tangent to the polar curve r = f (θ) is
=
dx
dy
dθ
dx
dθ
==
dr
dθ
dr
dθ
sin θ + r cos θ
cos θ − r sin θ
10.4
Conic Sections.
1. conics = set of points that result from intersecting a cone with a plane: parabolas, ellipses, hyperbolas
2. an equation of the parabola with focus (0, p) and directrix y = −p (focus =
fixed point, and directrix = a fixed line) is
x2 = 4py
(or y 2 = 4px if the focus is (p, 0) and the directrix x = −p)
3. an equation of the ellipse with foci (±c, 0) (where c2 = a2 − b2 , a ≥ b > 0) and
vertices (±a, 0) is
x2 y 2
+ 2 =1
a2
b
4. an equation of the hyperbola with focii (±c, 0) ((where c2 = a2 + b2 ), vertices
is
(±a, 0), and asymptotes y = ± bx
a
x2 y 2
− 2 =1
a2
b
5. an equation of the hyperbola with focii (0, ±c) ((where c2 = a2 + b2 ), vertices
is
(0, ±a), and asymptotes y = ± ax
b
y 2 x2
− 2 =1
a2
b