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

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

Chapter 9, 9.2, P491 if

Chapter 9, 9.2, P491

Note that

Chapter 9, 9.2, P 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 α to β, then the length of C is

Chapter 9, 9.3, P498 The point P is represented by the ordered pair (r, Θ) and r, Θ are called polar coordinates of P. Polar coordinates system

Chapter 9, 9.3, P498

Chapter 9, 9.3, P499

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, P500 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, 9.4, P507 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, P509 The length of a curve with polar equation r=f( Θ), a≤ Θ ≤b, is

Chapter 9, 9.5, P511 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

Chapter 9, 9.5, P512 An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. These two fixed points are called the foci (plural of focus.)

Chapter 9, 9.5, P512

1.The ellipse has foci(± c,0), where c 2 =a 2 -b 2,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 F 1 and F 2 (the foci) is a constant.

Chapter 9, 9.5, P The hyperbola has foci(± c,0), where c 2 =a 2+ b 2, vertices (± a,0), and asymptotes y=±(b/a)x.

Chapter 9, 9.5, P513 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, P514

8. THEOREM A polar equation of the form or represents a conic section with eccentricity e. The conic is an ellipse if e 1.