Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations
Section 10.1 Geometry of Parabola, Ellipse, Hyperbola Geometric Definition Parabola Ellipse Hyperbola Translations Distance Between a Point and Line Parabolic Mirrors Optical Consequences Elliptical Reflectors Hyperbolic Reflectors Section 10.2 Polar Coordinates Illustrative Figure Assigning Polar Coordinates Properties 1 and 2 Property 3 Relation to Rectangular Coordinates Properties Relating Polar and Rectangular Coordinates Simple Sets Symmetry Section 10.3 Sketching Curves in Polar Coordinates Spiral of Archimedes Example Lines Circles Limaçons Lemniscates Petal Curves Intersection of Polar Curves Section 10.4 Area in Polar Coordinates Computing Area Properties Section 10.5 Curves Given Parametrically Parameterized Curve Straight Lines Ellipses and Circles Hyperbolas Section 10.6 Tangents to Curves Given Parametrically Assumptions Section 10.7 Arc Length and Speed Length of a Curve Formula Length of the Graph of f Geometric Significance of dx/ds and dy/ds 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 The Area of a Surface of Revolution Centroid of a Curve Formulas Pappus’s Theorem on Surface Area Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Geometry Of Parabolas Geometric Definition
Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Geometry Of Parabolas Parabola
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 to x2 = 4cy. Terminology A parabola has a focus, a directrix, a vertex, and an axis. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Geometry Of Hyperbolas
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 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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Geometry Of Parabola, Ellipse, Hyperbola
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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Polar Coordinates Polar 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: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Polar Coordinates (3) Adding π to the second coordinate is equivalent to changing the sign of the first coordinate: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Polar Coordinates Relation to Rectangular Coordinates
The relation between polar coordinates [r, θ] and rectangular coordinates (x, y) is given by the following equations: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 equation r = 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 equation r cos θ = a or, equivalently, r = a sec θ (4) For b ≠ 0, the horizontal line y = b has polar equation r sin θ = b or, equivalently, r = b csc θ. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Polar Coordinates Symmetry

Sketching Curves in Polar Coordinates
Example Sketch the curve r = θ, θ ≥ 0 in polar coordinates. Solution At θ = 0, r = 0; at θ = ¼π, r = ¼ π; at θ = ½π, r = ½ π; and so on. The curve is shown in detail from θ = 0 to θ = 2π in Figure 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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Example Sketch the curve r = cos 2θ in polar coordinates. Solution Since 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 into polar coordinates [r, θ], we obtain a sketch of the curve r = cos 2θ in polar coordinates (Figure ). The sketch is developed in eight stages. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Intersection of Polar Curves The 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 = but 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, 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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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-coordinate x(t) and y-coordinate y(t). Then, as t ranges over I, the point (x(t), y(t)) traces out a path in the xy-plane. We call such a path a parametrized curve and refer to t as the parameter. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Straight Lines Given that (x0, y0) = (x1, y1), the functions parametrize the line that passes through the points (x0, y0) and (x1, y1). Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 circle x2 + y2 = a2 by setting Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Tangents to Curves Given Parametrically
Let C be a curve parametrized by the functions x = 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 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Tangents to Curves Given Parametrically

Arc Length and Speed Figure represents a curve C parametrized by a pair of functions x = 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. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Arc Length and Speed 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 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Arc Length and Speed Suppose now that C is the graph of a continuously differentiable function y = f (x), x  [a, b]. We can parametrize C by setting x(t) = t, y(t) = f (t) t  [a, b]. Since x´(t) = and y´(t) = f´(t), (10.7.1) gives Replacing t by x, we can write: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Arc Length and Speed The Geometric Significance of dx/ds and dy/ds

Arc Length and Speed Speed Along a Plane Curve
So 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 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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

We can locate the centroid of a curve from the following principles, which we take from 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 Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

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Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

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