Chapter 10: Conic Sections; Polar Coordinates; Parametric ...

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Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Section 10.1 Geometry of Parabola, Ellipse, Hyperbolaa. Geometric Definitionb. Parabolac. Ellipsed. Hyperbolae. Translationsf. Distance Between a Point and Lineg. Parabolic Mirrorsh. Optical Consequencesi. Elliptical Reflectorsj. Hyperbolic Reflectors

Section 10.2 Polar Coordinatesa. Illustrative Figureb. Assigning Polar Coordinatesc. Properties 1 and 2d. Property 3e. Relation to Rectangular Coordinatesf. Properties Relating Polar and Rectangular Coordinatesg. Simple Setsh. Symmetry

Section 10.3 Sketching Curves in Polar Coordinatesa. Spiral of Archimedesb. Examplec. Linesd. Circlese. Limaçonsf. Lemniscatesg. Petal Curvesh. Intersection of Polar Curves

Section 10.4 Area in Polar Coordinatesa. Computing Areab. Properties

Section 10.5 Curves Given Parametricallya. Parameterized Curveb. Straight Linesc. Ellipses and Circlesd. Hyperbolas

Section 10.6 Tangents to Curves Given Parametricallya. Assumptionsb. Properties

Section 10.7 Arc Length and Speeda. Length of a Curveb. Formulac. Length of the Graph of fd. Geometric Significance of dx/ds and dy/dse. Speed Along a Plane Curve

Section 10.7 The Area of a Surface of Revolution; The Centroid of a Curve; Pappus’s Theorem on Surface Area

a. The Area of a Surface of Revolution b. Computing Areac. Centroid of a Curved. Formulase. Pappus’s Theorem on Surface Area



Geometry Of Parabolas

Geometric Definition



Geometry Of ParabolasParabola

Standard Position F on the positive y-axis, l horizontal. Then F has coordinates of the form (c, 0) with c > 0 and l has equation x = −c.

Derivation of the Equation A point P(x, y) lies on the parabola iff d1 = d2, which here means

This equation simplifies tox2 = 4cy.

Terminology A parabola has a focus, a directrix, a vertex, and an axis.

( )22x y c y c+ − = +



Geometry Of Ellipses

Ellipse

Standard Position F1 and F2 on the x-axis at equal distances c from the origin. Then F1 is at (−c, 0) and F2 at (c, 0). With d1 and d2 as in the defining figure, set d1 + d2 = 2a.

Equation

Setting , we have

2 2

2 2 2 1x ya a c

+ =−

2 2b a c= −2 2

2 2 1x ya b

+ =

Terminology An ellipse has two foci, F1 and F2, a major axis, a minor axis, and four vertices. The point at which the axes of the ellipse intersect is called the center of the ellipse.



Geometry Of Hyperbolas

Hyperbola

Standard Position F1 and F2 on the x-axis at equal distances c from the origin.Then F1 is at (−c, 0) and F2 at (c, 0). With d1 and d2 as in the defining figure, set |d1 − d2| = 2a

Equation

Terminology A hyperbola has two foci, F1 and F2, two vertices, a transverse axis that joins the two vertices, and two asymptotes. The midpoint of the transverse axis is called the center of the hyperbola.

Setting , we have

2 2

2 2 2 1x ya c a

− =−

2 2b c a= −2 2

2 2 1x ya b

− =



Translations

Suppose that x0 and y0 are real numbers and S is a set in the xy-plane. By replacing each point (x, y) of S by (x + x0, y + y0), we obtain a set S´ which is congruent to S and obtained from S without any rotation. Such a displacement is called a translation.

The translation(x, y) → (x + x0, y + y0)

applied to a curve C with equation E(x, y) = 0 results in a curve C´ with equation E(x − x0, y − y0) = 0.

Geometry Of Parabola, Ellipse, Hyperbola



The distance between the origin and any line l : Ax + By + C = 0 is given by the formula

By means of a translation we can show that the distance between any point P(x0, y0) and the line l : Ax + By + C = 0 is given by the formula

( )2 2

0,C

d lA B

=+




Parabolic Mirrors

Take a parabola and revolve it about its axis. This gives you a parabolic surface. A curved mirror of this form is called a parabolic mirror. Such mirrors are used in searchlights (automotive headlights, flashlights, etc.) and in reflecting telescopes.







Elliptical Reflectors




Hyperbolic Reflectors




Polar Coordinates



Polar Coordinates



Polar CoordinatesPolar coordinates are not unique. Many pairs [r, θ] can represent the same point.

(1) If r = 0, it does not matter how we choose θ. The resulting point is still the pole:

(2) Geometrically there is no distinction between angles that differ by an integer multiple of 2π. Consequently:



Polar Coordinates

(3) Adding π to the second coordinate is equivalent to changing the sign of the first coordinate:



Polar Coordinates

Relation to Rectangular Coordinates

The relation between polar coordinates [r, θ] and rectangular coordinates (x, y) is given by the following equations:



Polar Coordinates

Unless x = 0,

and, under all circumstances,



Polar Coordinates

Here are some simple sets specified in polar coordinates.

(1) The circle of radius a centered at the origin is given by the equationr = a.

The interior of the circle is given by r < a and the exterior by r > a.

(2) The line that passes through the origin with an inclination of α radians has polar equation

θ = α.

(3) For a ≠ 0, the vertical line x = a has polar equationr cos θ = a or, equivalently, r = a sec θ

(4) For b ≠ 0, the horizontal line y = b has polar equationr sin θ = b or, equivalently, r = b csc θ.



Polar Coordinates

Symmetry



Sketching Curves in Polar Coordinates

ExampleSketch the curve r = θ, θ ≥ 0 in polar coordinates.

SolutionAt θ = 0, r = 0; at θ = ¼π, r = ¼ π; at θ = ½π, r = ½ π; and so on. Thecurve is shown in detail from θ = 0 to θ = 2π in Figure 10.3.1. It is an unending spiral, the spiral of Archimedes. More of the spiral is shown on a smaller scale in the right part of the figure.



Sketching Curves in Polar CoordinatesExampleSketch the curve r = cos 2θ in polar coordinates.SolutionSince the cosine function has period 2π, the function r = cos 2θ has period π. Thus it may seem that we can restrict ourselves to sketching the curve from θ = 0 to θ = π. But this is not the case. To obtain the complete curve, we must account for r in every direction; that is, from θ = 0 to θ = 2π.

Translating Figure 10.3.4 into polar coordinates [r, θ], we obtain a sketch of the curve r = cos 2θ in polar coordinates (Figure 10.3.5). The sketch is developed in eight stages.




Lines : θ = a, r = a sec θ, r = a csc θ.




Circles : r = a, r = a sin θ, r = a cos θ.




Limaçons : r = a + b sin θ, r = a + b cos θ.




Lemniscates: r² = a sin 2θ, r² = a cos 2θ




Petal Curves: r = a sin nθ, r = a cos nθ, integer n.

If n is odd, there are n petals. If n is even, there are 2n petals.




The Intersection of Polar CurvesThe fact that a single point has many pairs of polar coordinates can cause complications. In particular, it means that a point [r1, θ1] can lie on a curve given by a polar equation although the coordinates r1 and θ1 do not satisfy the equation. For example, the coordinates of [2, π] do not satisfy the equation r2 = 4 cos θ:

r2 = 22 = 4 but 4 cos θ = 4 cos π = −4.

Nevertheless the point [2, π] does lie on the curve r2 = 4 cos θ. We know this because [2, π] = [−2, 0] and the coordinates of [−2, 0] do satisfy the equation:

r2 = (−2)2 = 4, 4 cos θ = 4 cos 0 = 4

In general, a point P[r1, θ1] lies on a curve given by a polar equation if it has at least one polar coordinate representation [r, θ] with coordinates that satisfy the equation. The difficulties are compounded when we deal with two or more curves.



Area in Polar Coordinates



Area in Polar Coordinates



Curves given Parametrically

Assume a pair of functions x = x(t), y = y(t) is differentiable on the interior of an interval I. At the endpoints of I (if any) we require only one-sided continuity.For each number t in I we can interpret (x(t), y(t)) as the point with x-coordinatex(t) and y-coordinate y(t). Then, as t ranges over I, the point (x(t), y(t)) traces out apath in the xy-plane. We call such a path a parametrized curve and refer to t as the parameter.




Straight Lines

Given that (x0, y0) = (x1, y1), the functions

parametrize the line that passes through the points (x0, y0) and (x1, y1).




Ellipses and Circles

Usually we let t range from 0 to 2π and parametrize the ellipse by setting

If b = a, we have a circle. We can parametrize the circlex2 + y2 = a2

by setting




Hyperbolas

Take a, b > 0. The functions x(t) = a cosh t, y(t) = b sinh t satisfy the identity

Since x(t) = a cosh t > 0 for all t, as t ranges over the set of real numbers, the point(x(t), y(t)) traces out the right branch of the hyperbola

( ) ( )2 2

2 2 1x t y t

a b − =

2 2

2 2 1x ya b

− =



Tangents to Curves Given Parametrically

Let C be a curve parametrized by the functionsx = x(t), y = y(t)

defined on some interval I. We will assume that I is an open interval and the parametrizing functions are differentiable.

Since a parametrized curve can intersect itself, at a point of C there can be

(i) one tangent, (ii) two or more tangents, or (iii) no tangent at all.

To make sure that there is at least one tangent line at each point of C, we will make the additional assumption that



Tangents to Curves Given Parametrically



Arc Length and Speed

Figure 10.7.1 represents a curve C parametrized by a pair of functionsx = x(t), y = y(t) t ∈ [a, b].

We will assume that the functions are continuously differentiable on [a, b] (have first derivatives which are continuous on [a, b]). We want to determine the length of C.




The length of the path C traced out by a pair of continuously differentiable functions

x = x(t), y = y(t) t ∈ [a, b]

is given by the formula




Suppose now that C is the graph of a continuously differentiable functiony = f (x), x ∈ [a, b].

We can parametrize C by settingx(t) = t, y(t) = f (t) t ∈ [a, b].

Since

x´(t) = 1 and y´(t) = f´(t),

(10.7.1) gives

( ) ( ) 21

b

aL C f t dt′= + ∫

Replacing t by x, we can write:




The Geometric Significance of dx/ds and dy/ds




Speed Along a Plane CurveSo far we have talked about speed only in connection with straight-line motion. How can we calculate the speed of an object that moves along a curve? Imagine an object moving along some curved path. Suppose that (x(t), y(t)) gives the position of the object at time t. The distance traveled by the object from time zero to any later time t is simply the length of the path up to time t:

The time rate of change of this distance is what we call the speed of the object. Denoting the speed of the object at time t by ν(t), we have

( ) ( ) ( )2 2

0

ts t x u y u du′ ′= + ∫



The Area Of A Surface Of Revolution; The CentroidOf A Curve; Pappus's Theorem On Surface Area

The Area of a Surface of Revolution







Centroid of a CurveWe can locate the centroid of a curve from the following principles, which we takefrom physics.

Principle 1: Symmetry. If a curve has an axis of symmetry, then the centroid lies somewhere along that axis.

Principle 2: Additivity. If a curve with length L is broken up into a finite number of pieces with arc lengths Δs1, . . . , Δsn and centroids then

( ),x y

( ) ( )1 1, , , , ,n nx y x y

1 1 1 1andn n n nxL x s x s yL y s y s= ∆ + + ∆ = ∆ + + ∆







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