ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates

ESSENTIAL CALCULUSESSENTIAL CALCULUS

CH09 Parametric CH09 Parametric equations and polar equations and polar

coordinatescoordinates

In this Chapter:In this Chapter:

9.1 Parametric Curves

9.2 Calculus with Parametric Curves

9.3 Polar Coordinates

9.4 Areas and Lengths in Polar Coordinates

9.5 Conic Sections in Polar Coordinates

Review

Suppose that x and y are both given as functions of a third variable t (called aparameter) by the equations

x=f (t) y=g (t)

(called parametric equations). Each value of t determines a point (x.y), which we can plot in a coordinate plane. As t varies, the point (x,y)=(f(t) . g(t)), varies and traces out a curve C, which we call a parametric curve.

Chapter 9, 9.1, P484


dtdxdtdy

dx

dy 0

dt

dxif


dtdxdxdy

dtd

dx

dy

dx

d

dx

yd )()(

2

2


2

2

2

2

2

2

dtxd

dtyd

dx

ydNote that


5. THEOREM If 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] and C is traversed exactly once as t increases from αtoβ , then the length of C is

dtdt

dy

dt

dxL

22 )()(


The point P is represented by the ordered pair (r,Θ) and r, Θ are called polar coordinates of P.

Polar coordinates system




If the point P has Cartesian coordinates (x,y) and polar coordinates (r,Θ), then, from the figure, we have

and so

1.

2.

r

xcos

r

ysin

cosrx sinry

222 yxr x

ytan


The graph of a polar equation r=f(Θ) , or more generally F (r,Θ)=0, consists of all points P that have at least one polar representation (r,Θ) whose coordinates satisfy the equation


The area A of the polar region R is

3.

Formula 3 is often written as

4.

with the understanding that r=f(Θ).

dfAb

a 2)]([2

1

drAb

a

2

2

1


The length of a curve with polar equation r=f(Θ), a≤Θ≤b , is

d

d

drrL

b

a 22 )(


A parabola is the set of points in a plane that are equidistant from a fixed point F (called the focus) and a fixed line (called the directrix). This definition is illustrated by Figure 1. Notice that the point halfway between the focus and the directrix lies on the parabola; it is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola.






An ellipse is the set of points in a plane the sum of whose distances from two fixedpoints F1 and F2 is a constant. These two fixed points are called the foci (plural of focus.)


12

2

2

2

b

y

a

x


12

2

2

2

b

y

a

x


12

2

2

2

b

y

a

x


1.The ellipse

has foci(± c,0), where c2=a2-b2 ,and vertices (± a,0),

12

2

2

2

b

y

a

x0ba


A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a constant.


2. The hyperbola

has foci(± c,0), where c2=a2+b2, vertices (± a,0), and asymptotes y=±(b/a)x.

12

2

2

2

b

y

a

x


3.THEOREM Let F be a fixed point (called the focus) and I be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called the eccentricity). The set of all points P in the plane such that

(that is, the ratio of the distance from F to the distance from I is the constant e) is a conic section. The conic is (a) an ellipse if e<1 (b) a parabola if e=1 (C) a hyperbola if e>1

ePl

PF


cos1)(

e

edra


cos1)(

e

edrb


sin1)(

e

edrc


sin1)(

e

edrd


8. THEOREM A polar equation of the form

or

represents a conic section with eccentricity e. The conic is an ellipse if e<1, a parabola if e=1, or a hyperbola if e>1.

cos1 e

edr

sin1 e

edr

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ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates