ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates

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ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates. 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 .

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### ESSENTIAL CALCULUSCH09 Parametric equations and polar coordinates

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 a

parameter) 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

if

Chapter 9, 9.2, P491

Note that

Chapter 9, 9.2, P491

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

Chapter 9, 9.2, P494

Polar coordinates system

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

Chapter 9, 9.3, P498

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

and so

1.

2.

Chapter 9, 9.3, P499

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

Chapter 9, 9.3, P500

The area A of the polar region R is

3.

Formula 3 is often written as

4.

with the understanding that r=f(Θ).

Chapter 9, 9.4, P507

The length of a curve with polar equation r=f(Θ)

, a≤Θ≤b , is

Chapter 9, 9.4, P509

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.

Chapter 9, 9.5, P511

An ellipse is the set of points in a plane the sum of whose distances from two fixed

points F1 and F2 is a constant. These two fixed points are called the foci (plural of focus.)

Chapter 9, 9.5, P512

The ellipse

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

Chapter 9, 9.5, P512

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.

Chapter 9, 9.5, P512

2. The hyperbola

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

Chapter 9, 9.5, P512

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

Chapter 9, 9.5, P513

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.

Chapter 9, 9.5, P514