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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

### Section 10.1 Geometry of Parabola, Ellipse, Hyperbola

### a. Geometric Definition

### b. Parabola

### c. Ellipse

### d. Hyperbola

### e. Translations

### f. Distance Between a Point and Line

### g. Parabolic Mirrors

### h. Optical Consequences

### i. Elliptical Reflectors

### j. Hyperbolic Reflectors Section 10.2 Polar Coordinates

### a. Illustrative Figure

### b. Assigning Polar Coordinates

### c. Properties 1 and 2

### d. Property 3

### e. Relation to Rectangular Coordinates

### f. Properties Relating Polar and Rectangular Coordinates

### g. Simple Sets

### h. Symmetry

### Section 10.3 Sketching Curves in Polar Coordinates

### a. Spiral of Archimedes

### b. Example

### c. Lines

### d. Circles

### e. Limaçons

### f. Lemniscates

### g. Petal Curves

### h. Intersection of Polar Curves

### Section 10.4 Area in Polar Coordinates

### a. Computing Area

### b. Properties

### Section 10.5 Curves Given Parametrically

### a. Parameterized Curve

### b. Straight Lines

### c. Ellipses and Circles

### d. Hyperbolas

### Section 10.6 Tangents to Curves Given Parametrically

### a. Assumptions

### b. Properties

### Section 10.7 Arc Length and Speed

### a. Length of a Curve

### b. Formula

### c. *Length of the Graph of f*

### d. *Geometric Significance of dx/ds and dy/ds*

### e. 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 Area

### c. Centroid of a Curve

### d. Formulas

### e. Pappus’s Theorem on Surface Area

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

## Geometry Of Parabolas

**Geometric Definition**

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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.*

**Standard Position F on the positive y-axis, l horizontal. Then F has coordinates**

**Derivation of the Equation A point P(x, y) lies on the parabola iff d** _{1} *= d* _{2} , which here means

**Derivation of the Equation A point P(x, y) lies on the parabola iff d**

### This equation simplifies to

*x* ^{2} *= 4cy.*

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

**Terminology A parabola has a focus, a directrix, a vertex,**

### ( ) ^{2}

*x* 2 + *y* − *c* = + *y* *c*

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## Geometry Of Ellipses

**Ellipse**

**Standard Position F** _{1} *and F* _{2} *on the x-axis at equal distances c from the origin. *

**Standard Position F**

*Then F* _{1} *is at (−c, 0) and F* _{2} *at (c, 0). With d* _{1} *and d* _{2} as in the defining figure, set *d* _{1} *+ d* _{2} *= 2a. *

**Equation**

### Setting , we have

### 2 2

### 2 2 2 1

*x* *y*

*a* + *a* *c* =

### −

### 2 2

*b* = *a* − *c*

### 2 2

### 2 2 1

*x* *y*

*a* + *b* =

**Terminology An ellipse has two foci, F** _{1} *and F* _{2} *, 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 *

**Terminology An ellipse has two foci, F**

### ellipse.

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## Geometry Of Hyperbolas

**Hyperbola**

**Standard Position F** _{1} *and F* _{2} *on the x-axis at equal distances c from the origin.*

**Standard Position F**

*Then F* _{1} *is at (−c, 0) and F* _{2} *at (c, 0). With d* _{1} *and d* _{2} as in the defining figure, set

*|d* _{1} *− d* _{2} *| = 2a* **Equation**

**Terminology A hyperbola has two foci, F** _{1} *and F* _{2} , *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.

**Terminology A hyperbola has two foci, F**

### Setting , we have

### 2 2

### 2 2 2 1

*x* *y*

*a* − *c* *a* =

### −

### 2 2

*b* = *c* − *a*

### 2 2

### 2 2 1

*x* *y*

*a* − *b* =

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**Translations**

*Suppose that x* _{0} *and y* _{0} *are real numbers and S is a set in the xy-plane. By * *replacing each point (x, y) of S by (x + x* _{0} *, y + y* _{0} *), 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 + x* _{0} *, y + y* _{0} )

*applied to a curve C with equation E(x, y) = 0 results in a curve C* ´ *with equation * *E(x * *− x* _{0} *, y * *− y* _{0} ) = 0.

## Geometry Of Parabola, Ellipse, Hyperbola

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*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(x* _{0} *, y* _{0} *) and the line l : Ax + By + C = 0 is given by the formula*

### ( ) 0, 2 *C* 2

*d* *l*

*A* *B*

### = +

## Geometry Of Parabola, Ellipse, Hyperbola

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**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.

## Geometry Of Parabola, Ellipse, Hyperbola

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## Geometry Of Parabola, Ellipse, Hyperbola

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**Elliptical Reflectors**

## Geometry Of Parabola, Ellipse, Hyperbola

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**Hyperbolic Reflectors**

## Geometry Of Parabola, Ellipse, Hyperbola

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## Polar Coordinates

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## Polar Coordinates

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## 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:

**(1) If r = 0, it does not matter how we choose**

**(2) Geometrically there is no distinction between angles that differ by an **

### integer multiple of 2 *π. Consequently:*

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## Polar Coordinates

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

### first coordinate:

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## Polar Coordinates

**Relation to Rectangular Coordinates**

### The relation between polar coordinates [ *r, θ] and rectangular coordinates *

*(x, y) is given by the following equations:*

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## Polar Coordinates

*Unless x = 0,*

### and, under all circumstances,

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## 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. *

**(1) The circle of radius a centered at the origin is given by the equation**

*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*

**(3) For a**

*r cos * *θ = a or, equivalently, r = a sec θ* **(4) For b ** *≠ 0, the horizontal line y = b has polar equation*

**(4) For b**

*r sin * *θ = b or, equivalently, r = b csc θ.*

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## Polar Coordinates

**Symmetry**

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## 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 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.

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## Sketching Curves in Polar Coordinates

**Example**

*Sketch the curve r = cos 2* *θ in polar coordinates.*

**Solution**

### Since the cosine function has period 2 *π, the function *

*r = cos 2* *θ 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.

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## Sketching Curves in Polar Coordinates

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

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## Sketching Curves in Polar Coordinates

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

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## Sketching Curves in Polar Coordinates

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

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## Sketching Curves in Polar Coordinates

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

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## Sketching Curves in Polar Coordinates

*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.*

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## Sketching Curves in Polar Coordinates

**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 [r* _{1} *, θ* 1 ] can lie on a curve given by *a polar equation although the coordinates r* _{1} and *θ* _{1} do not satisfy the equation. For example, the coordinates of [2 *, π] do not satisfy the equation r* ^{2} = 4 cos *θ:*

*r* ^{2} = 2 ^{2} = 4 but 4 cos *θ = 4 cos π = −4.*

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

*r* ^{2} = (−2) ^{2} *= 4, 4 cos * *θ = 4 cos 0 = 4*

*In general, a point P[r* _{1} *, θ* _{1} ] 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.

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## Area in Polar Coordinates

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## Area in Polar Coordinates

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## 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.*

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## Curves given Parametrically

**Straight Lines**

*Given that (x* _{0} *, y* _{0} *) = (x* _{1} *, y* _{1} ), the functions

*parametrize the line that passes through the points (x* _{0} *, y* _{0} *) and (x* _{1} *, y* _{1} ).

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## Curves given Parametrically

**Ellipses and Circles**

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

*If b = a, we have a circle. We can parametrize the circle* *x* ^{2} *+ y* ^{2} *= a* ^{2}

### by setting

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## Curves given Parametrically

**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 1

*x t* *y t*

*a* *b*

###

### − =

### 2 2

### 2 2 1

*x* *y*

*a* − *b* =

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## 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

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## Tangents to Curves Given Parametrically

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## Arc Length and Speed

*Figure 10.7.1 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.*

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## 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

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## 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) = 1 and y* ´ *(t) = f* ´ *(t),* (10.7.1) gives

### ( ) ^{b} ^{1} ( ) ^{2}

^{b}

*L C* = ∫ *a* + *f* ′ *t* *dt*

*Replacing t by x, we can write:*

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## Arc Length and Speed

**The Geometric Significance of dx/ds and dy/ds**

**The Geometric Significance of dx/ds and dy/ds**

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## 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*

### ( ) ( ) ^{2} ( ) ^{2}

### 0

*s t* = ∫ *t* *x u* ′ + *y u* ′ *du*

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### 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**

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### The Area Of A Surface Of Revolution; The Centroid

### Of A Curve; Pappus's Theorem On Surface Area

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

**Centroid of a Curve**

### 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 1: Symmetry. If a curve has an axis of symmetry, then the centroid lies**

**Principle 2: Additivity. If a curve with length L is broken up into a finite number of pieces with ** arc lengths *Δs* 1 *, . . . , * *Δs* *n* and centroids then

**Principle 2: Additivity. If a curve with length L is broken up into a finite number of pieces with**

### ( ^{x y} ^{,} )

^{x y}

### ( *x y* 1 , 1 ) , , ( *x y* _{n} , _{n} ) ,

_{n}

_{n}

### 1 1 _{n} _{n} and 1 1 _{n} _{n}

_{n}

_{n}

_{n}

_{n}