Parametric, Vector and Polar Function

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Parametric, Vector and Polar Function

• Dr. Ching I Chen

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10.1 Parametric Functions (1)Derivative

• If f(t) and g(t) are functions of t, then the
  curve given by the parametric equations x f(t)
  , y g(t) can be treated as the graph of a
  function of the parameter t.

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10.1 Parametric Functions (2)Derivative
When a body travels in x-y plane, the parametric
equations can be used to model the bodys
motion and path.
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10.1 Parametric Functions (3)Derivative
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10.1 Parametric Functions (4, Example
1)Derivative
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10.1 Parametric Functions (5) Derivatives d2y/dx2
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10.1 Parametric Functions (6, Example 2)
Derivatives d2y/dx2

• Example 2 Find d2y/dx2 as a function t, if x t
  - t2,
• y t - t3

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10.1 Parametric Functions (7) Length of a Smooth
Curve
x
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10.1 Parametric Functions (8, Example 3) Length
of a Smooth Curve

• Example 3 Find the length of the astroid, if x
  cos3 t
• y sin3 t

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10.1 Parametric Functions (9, Example 4)Cycloids
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10.1 Parametric Functions (10)Cycloids
(Exploration 1-1)
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10.1 Parametric Functions (11)Cycloids
(Exploration 1-2)
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10.1 Parametric Functions (12, Example 5)Cycloids
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10.1 Parametric Functions (13)Surface Area
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10.1 Parametric Functions (14)Surface Area

• Geometric interpretation

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10.1 Parametric Functions (15)Surface Area

• Geometric interpretation

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10.1 Parametric Functions (16, Example 6)Surface
Area
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10.1 Parametric Functions (17)Surface Area
(Exercises 29)
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10.2 Vectors in the Plane (1) Component Form

• Scalar a real number to describe the quantity
  of the physical terminology, such as mass,
  length, density.
• Vector not only a real number but also
  direction to describe the quantity of the
  physical terminology, such as position, velocity,
  force.

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10.2 Vectors in the Plane (2) Component Form

• Directed line

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10.2 Vectors in the Plane (3, Example 1)
Component Form
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10.2 Vectors in the Plane (4) Component Form

• Vector form
• unit vector form v a i b j
• component form v lta, bgt

The value vx abd vy are the component of v. The
magnitude (length) of v is
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10.2 Vectors in the Plane (5, Example 2)
Component Form
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10.2 Vectors in the Plane (6) Component Form

• Vector form

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10.2 Vectors in the Plane (7, Example 3)
Component Form
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10.2 Vectors in the Plane (8) Component Form
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10.2 Vectors in the Plane (9) Vector Operations
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10.2 Vectors in the Plane (10, Example 4) Vector
Operations
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10.2 Vectors in the Plane (11) Vector Operations
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10.2 Vectors in the Plane (12, Theorem 1) Angle
Between Vectors
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10.2 Vectors in the Plane (13) Angle Between
Vectors
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10.2 Vectors in the Plane (14, Example 5) Angle
Between Vectors
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10.2 Vectors in the Plane (15, Example 6)
Applications
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10.2 Vectors in the Plane (16, Example 6)
Applications
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10.3 Vector-valued Functions (1) Standard Unit
Vectors

• Any vector v a i b j can be considered as a
  linear
• combination of two standard unit vectors
• i ?1, 0 ? and j ?0, 1 ?
• This is so called unit vector form of a vector
• v ?a, b ? ?a, 0 ? ?0, b ?
• a?1, 0 ? b?0, 1 ? a i b j

a horizontal (x) component b vertical (y)
component
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10.3 Vectors in the Plane (2, Example 1) Standard
Unit Vectors
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10.3 Vector-valued Functions (3) Planar Curves

• The component of a vector is a function (not a
  real number), namely
• r(t) f(t) i g(t) j

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10.3 Vector-valued Functions (4) Planar Curves
(Example 2)
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10.3 Vector-valued Functions (5) Planar Curves

• Exercise 10.3 (20) A curve function as follows
• r(t) sin t i t j, t gt 0

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10.3 Vector-valued Functions (6) Limits and
Continuity
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10.3 Vector-valued Functions (7) Limits and
Continuity (Example 3)
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10.3 Vector-valued Functions (8) Limits and
Continuity
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10.3 Vector-valued Functions (9) Limits and
Continuity (Example 4)
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10.3 Vector-valued Functions (10) Limits and
Continuity
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10.3 Vector-valued Functions (11) Derivatives
and Motion

• Suppose that r(t) f(t) i g(t) j is the
  position of a particle moving along a curve in
  the plane and that f(t) and g(t) are
  differentiable functions of t. Then the
  difference between the particles positions at
  time tDt and the time t is

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10.3 Vector-valued Functions (12) Derivatives
and Motion

• As Dt approaches zero, three things seem to
  happen simultaneously.
• Q approaches P along the curve.
• The secant line PQ seems to approach a limiting
  position tangent to the curve at P.
• The quotient Dr(t)/Dt approaches the limit

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10.3 Vector-valued Functions (13) Derivatives
and Motion
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10.3 Vector-valued Functions (14) Derivatives
and Motion

1. If dr/dt is continuous and never 0 for both
   component, the curve traced by r is smooth, there
   are no sharp corners or cusps.
2. The vector dr/dt when different from 0, is also a
   vector tangent to the curve. The tangent line to
   the curve at a point P (f(a), g(a)) is defined
   be the line through P parallel to dr/dt at t a.

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10.3 Vector-valued Functions (15) Derivatives
and Motion
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10.3 Vector-valued Functions (16) Derivatives
and Motion (Example 5-a,b)
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10.3 Vector-valued Functions (17) Derivatives
and Motion (Example 5-c)
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10.3 Vector-valued Functions (18) Derivatives
and Motion (Example 6-a)
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10.3 Vector-valued Functions (19) Derivatives
and Motion (Example 6-b )
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10.3 Vector-valued Functions (20)
Differentiation Rules
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10.3 Vector-valued Functions (21)
Differentiation Rules
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10.3 Vector-valued Functions (22) Integrals
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10.3 Vector-valued Functions (23) Integrals
(Example 7)
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10.3 Vector-valued Functions (24) Integrals
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10.3 Vector-valued Functions (25) Integrals
(Example 8)
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10.3 Vector-valued Functions (26) Integrals
(Example 9-a)
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10.3 Vector-valued Functions (26) Integrals
(Example 9-a)
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10.3 Vector-valued Functions (26) Integrals
(Example 9-b)
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10.4 Modeling Projection Motion (1)Ideal
Projection Motion

• A particle moving in a vertical plane
• Only gravity force acts on the particle

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10.4 Modeling Projection Motion (2)Ideal
Projection Motion

• Assumption
• The projectile is launched from the original at
  time t 0, with an initial velocity vo which
  make an angle with the horizontal.

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10.4 Modeling Projection Motion (3)Ideal
Projection Motion

• Equation of motion
• Newtons second law of motion the force acting
  on the projectile is equal to the projectiles
  mass times its acceleration

• Question
• Can we know the motion status e.g. position,
  velocity, acceleration etc.

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10.4 Modeling Projection Motion (4)Ideal
Projection Motion
This is ideal projectile motion with vector
equation. a launch angle, vo initial speed.
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10.4 Modeling Projection Motion (5)Ideal
Projection Motion
The vector equation also can be expressed as
parameter equations
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10.4 Modeling Projection Motion (5)Ideal
Projection Motion (Example 1)
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10.4 Modeling Projection Motion (7) Height,
Flight time, and Range
What is so called Flight time ?
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10.4 Modeling Projection Motion (8) Height,
Flight time, and Range
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10.4 Modeling Projection Motion (9) Height,
Flight time, and Range
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Modeling Projection Motion (10, Example2)
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10.4 Modeling Projection Motion (11) Ideal
Trajectories Are Parabolic

• General Consideration

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10.4 Modeling Projection Motion (12) Ideal
Trajectories Are Parabolic (Example 3)
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10.4 Modeling Projection Motion (13) Ideal
Trajectories Are Parabolic (Example 3-a)
?
ymax 74 ft
y
vo
a
6 ft
90 ft
x
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10.4 Modeling Projection Motion (14) Ideal
Trajectories Are Parabolic (Example 3-b)
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10.4 Modeling Projection Motion (15) Ideal
Trajectories Are Parabolic (Example 3-c)
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10.4 Modeling Projection Motion (16) Ideal
Trajectories Are Parabolic (Example 3-d)
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10.4 Modeling Projection Motion (17) Projectile
Motion with Wind Gusts (Example 4-a)
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10.4 Modeling Projection Motion (18) Projectile
Motion with Wind Gusts (Example 4-a)
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10.4 Modeling Projection Motion (19) Projectile
Motion with Wind Gusts (Example 4-b)
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10.4 Modeling Projection Motion (20) Projectile
Motion with Wind Gusts (Example 4-c)
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10.4 Modeling Projection Motion (21) Projectile
Motion with Wind Gusts (Exploration 1-1)
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10.4 Modeling Projection Motion (22) Projectile
Motion with Wind Gusts (Exploration 1-2)
y
vo152
8.8
a
3 ft
x
400 ft
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10.4 Modeling Projection Motion (23) Projectile
Motion with Wind Gusts (Exploration 1-3)
y
vo152
15 ft
3 ft
x
400 ft
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10.4 Modeling Projection Motion (24)Projectile
Motion with Air Resistance

• Drag Force
• Drag force is one of the most popular external
  force acting to the projectile.
• Drag force acts in a direction opposite to the
  velocity of the projectile.
• Low velocity, the drag force is proportional to
  the speed which is called linear.
• High speed, the drag force is proportional to
  different powers of the speed over different
  velocity ranges.

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10.4 Modeling Projection Motion (25)Projectile
Motion with Air Resistance

• Equation of motion

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10.4 Modeling Projection Motion (26)Projectile
Motion with Air Resistance

• Equation of motion

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10.4 Modeling Projection Motion (26)Projectile
Motion with Air Resistance

• Solving x direction

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10.4 Modeling Projection Motion (27)Projectile
Motion with Air Resistance

• Solving y direction

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10.4 Modeling Projection Motion (28)Projectile
Motion with Air Resistance

• Solving y direction

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10.4 Modeling Projection Motion (29)Projectile
Motion with Air Resistance (Exploration 2-1)
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10.4 Modeling Projection Motion (30)Projectile
Motion with Air Resistance (Exploration 2-1)
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10.4 Modeling Projection Motion (31)Projectile
Motion with Air Resistance (Exploration 2-2)
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10.4 Modeling Projection Motion (32)Projectile
Motion with Air Resistance (Exploration 2-3)
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10.5 Polar Coordinates and Polar Graphs (1)
Polar Coordinates
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10.5 Polar Coordinates and Polar Graphs (2)
Polar Coordinates (Example 1)
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10.5 Polar Coordinates and Polar Graphs (3)
Polar Graphing
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10.5 Polar Coordinates and Polar Graphs (4)
Polar Graphing (Example 2)
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10.5 Polar Coordinates and Polar Graphs (5)
Polar Graphing (Example 3)
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10.5 Polar Coordinates and Polar Graphs (6)
Polar Graphing

• Symmetry

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10.5 Polar Coordinates and Polar Graphs (7)
Polar Graphing (Exploration 1)
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10.5 Polar Coordinates and Polar Graphs (8)
Relating Polar and Cartesian Coordinates
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10.5 Polar Coordinates and Polar Graphs (9)
Relating Polar and Cartesian Coordinates (Example
4)
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10.5 Polar Coordinates and Polar Graphs (10)
Relating Polar and Cartesian Coordinates (Example
5)
What is the graph for r 0
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10.5 Polar Coordinates and Polar Graphs (11)
Relating Polar and Cartesian Coordinates (Example
6)
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10.5 Polar Coordinates and Polar Graphs (12)
Relating Polar and Cartesian Coordinates (Exp. 2)
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10.6 Calculus of Polar Curve (1)Slope

• A function y f(x), one may find its derivative
  by dy/dx
• Any Cartesian function is equivalent to polar
  system.
• How to find the slope in term of polar coordinate
  ?

dy/dx
y
Function y f(x)
Function r f (q)
x
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10.6 Calculus of Polar Curve (2)Slope

• Since in polar coordinate r f(q)
• One expresses x r cosq f(q)cos(q)

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10.6 Calculus of Polar Curve (3)Slope
horizontal tangent
y
vertical tangent
x
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10.6 Calculus of Polar Curve (4)Slope (Example
1-a)
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10.6 Calculus of Polar Curve (5)Slope (Example
1-b)
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10.6 Calculus of Polar Curve (6)Slope (Example 1)
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10.6 Calculus of Polar Curve (7)Slope (Example 2)
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10.6 Calculus of Polar Curve (8)Slope (Example 2)
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10.6 Calculus of Polar Curve (9)Area in the Plane

• The region OTS is bounded by the rays q a and
  q b and curve r f(q).

O
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10.6 Calculus of Polar Curve (10)Area in the
Plane
O
x
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10.6 Calculus of Polar Curve (11)Area in the
Plane (Example 3)
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10.6 Calculus of Polar Curve (12)Area in the
Plane (Example 4)
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10.6 Calculus of Polar Curve (13)Area in the
Plane

• Area Between Polar Curve

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10.6 Calculus of Polar Curve (14)Area in the
Plane (Example 5)
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10.6 Calculus of Polar Curve (15) Length of a
Curve
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10.6 Calculus of Polar Curve (16) Length of a
Curve (Example 6)
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10.6 Calculus of Polar Curve (17)Area of a
Surface of Revolution
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10.6 Calculus of Polar Curve (18)Area of a
Surface of Revolution
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10.6 Calculus of Polar Curve (19)Area of a
Surface of Revolution (Example 7)