PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this...

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PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATES AND POLAR COORDINATES 10

Transcript of PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this...

Page 1: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATESAND POLAR COORDINATES

10

Page 2: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

10.4Areas and Lengths

in Polar Coordinates

In this section, we will:

Develop the formula for the area of a region

whose boundary is given by a polar equation.

PARAMETRIC EQUATIONS & POLAR COORDINATES

Page 3: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

We need to use the formula for the area

of a sector of a circle

A = ½r2θ

where:

r is the radius. θ is the radian measure

of the central angle.

Formula 1

Page 4: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Formula 1 follows from the fact that

the area of a sector is proportional to

its central angle:

A = (θ/2π)πr2 = ½r2θ

Page 5: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Let R be the region bounded by the polar

curve r = f(θ) and by the rays θ = a and θ = b,

where:

f is a positive continuous function.

0 < b – a ≤ 2π

Page 6: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

We divide the interval [a, b] into subintervals

with endpoints θ0, θ1, θ2, …, θn, and equal

width ∆θ.

Then, the rays θ = θi divide R into smaller regions with central angle ∆θ = θi – θi–1.

Page 7: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

If we choose θi* in the i th subinterval [θi–1, θi]

then the area ∆Ai of the i th region is the area

of the sector of a circle with central angle ∆θ

and radius f(θ*).

Page 8: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Thus, from Formula 1, we have:

∆Ai ≈ ½[f(θi*)]2 ∆θ

So, an approximation to the total area

A of R is: * 21

21

[ ( )]n

ii

A f

Formula 2

Page 9: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

It appears that the approximation

in Formula 2 improves as n → ∞.

Page 10: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

However, the sums in Formula 2 are

Riemann sums for the function g(θ) = ½[f(θ)]2.

So, * 2 21 1

2 21

lim [ ( )] [ ( )]n b

i ani

f f d

Page 11: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Therefore, it appears plausible—and can, in

fact, be proved—that the formula for the area

A of the polar region R is:

212 [ ( )]

b

aA f d

Formula 3

Page 12: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Formula 3 is often written as

with the understanding that r = f(θ).

Note the similarity between Formulas 1 and 4.

212

b

aA r d

Formula 4

Page 13: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

When we apply Formula 3 or 4, it is helpful

to think of the area as being swept out by

a rotating ray through O that starts with

angle a and ends with angle b.

Formula 4

Page 14: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Find the area enclosed by one loop of

the four-leaved rose r = cos 2θ.

The curve r = cos 2θ was sketched in Example 8 in Section 10.3

Example 1

Page 15: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Notice that the region enclosed by the right

loop is swept out by a ray that rotates from

θ = –π/4 to θ = π/4.

Example 1

Page 16: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Hence, Formula 4 gives:

4 2124

4 212 4

4 2

0

4120

41 12 4 0

cos 2

cos 2

(1 cos 4 )

sin 48

A r d

d

d

d

Example 1

Page 17: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Find the area of the region that lies

inside the circle r = 3 sin θ and outside

the cardioid r = 1 + sin θ.

Example 2

Page 18: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

The values of a and b in Formula 4

are determined by finding the points

of intersection of the two curves.

Example 2

Page 19: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

They intersect when 3 sin θ = 1 + sin θ,

which gives sin θ = ½.

So, θ = π/6 and 5π/6.

Example 2

Page 20: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

The desired area can be found by subtracting

the area inside the cardioid between θ = π/6

and θ = 5π/6 from the area inside the circle

from π/6 to 5π/6.

Example 2

Page 21: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Thus,

5 6 212 6

5 6 212 6

(3sin )

(1 sin )

A d

d

Example 2

Page 22: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

As the region is symmetric about the vertical

axis θ = π/2, we can write:

2 22 21 12 26 6

2 2

6

2 2 126

2

6

2 9sin (1 2sin sin )

(8sin 1 2sin )

(3 4cos 2 2sin ) assin (1 cos 2 )

3 2sin 2 2cos

A d d

d

d

Example 2

Page 23: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

Example 2 illustrates the procedure for

finding the area of the region bounded by

two polar curves.

Page 24: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

In general, let R be a region that is

bounded by curves with polar equations

r = f(θ), r = g(θ), θ = a, θ = b,

where: f(θ) ≥ g(θ) ≥ 0 0 < b – a < 2π

Page 25: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

The area A of R is found by subtracting

the area inside r = g(θ) from the area inside

r = f(θ).

Page 26: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

AREAS IN POLAR COORDINATES

So, using Formula 3, we have:

2 2

2 2

1 1[ ( )] [ ( )]2 2

1[ ( )] [ ( )]

2

b b

a a

b

a

A f d g d

f g d

Page 27: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

The fact that a single point has many

representations in polar coordinates

sometimes makes it difficult to find all

the points of intersection of two polar curves.

Page 28: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

For instance, it is obvious from this figure

that the circle and the cardioid have three

points of intersection.

Page 29: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

However, in Example 2, we solved

the equations r = 3 sin θ and r = 1 + sin θ

and found only two such points:

(3/2, π/6) and (3/2, 5π/6)

Page 30: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

The origin is also a point of intersection.

However, we can’t find it by solving

the equations of the curves.

The origin has no single representation in polar coordinates that satisfies both equations.

Page 31: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

Notice that, when represented as

(0, 0) or (0, π), the origin satisfies

r = 3 sin θ.

So, it lies on the circle.

Page 32: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

When represented as (0, 3 π/2),

it satisfies r = 1 + sin θ.

So, it lies on the cardioid.

Page 33: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

Think of two points moving along

the curves as the parameter value θ

increases from 0 to 2π.

On one curve, the origin is reached at θ = 0 and θ = π.

On the other, it is reached at θ = 3π/2.

Page 34: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

The points don’t collide at the origin since they reach the origin at different times.

However, the curves intersect there nonetheless.

Page 35: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

CAUTION

Thus, to find all points of intersection of two

polar curves, it is recommended that you

draw the graphs of both curves.

It is especially convenient to use a graphing calculator or computer to help with this task.

Page 36: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

POINTS OF INTERSECTION

Find all points of intersection of

the curves r = cos 2θ and r = ½.

If we solve the equations r = cos 2θ and r = ½, we get cos 2θ = ½.

Therefore, 2θ = π/3, 5π/3, 7π/3, 11π/3.

Example 3

Page 37: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

POINTS OF INTERSECTION

Thus, the values of θ between 0 and 2π that satisfy both equations are:

θ = π/6, 5π/6, 7π/6, 11π/6

Example 3

Page 38: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

POINTS OF INTERSECTION

We have found four points

of intersection:

(½, π/6), (½, 5π/6), (½, 7π/6), (½, 11π/6)

Example 3

Page 39: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

POINTS OF INTERSECTION

However, you can see that the curves have

four other points of intersection:

(½, π/3), (½, 2π/3), (½, 4π/3), (½, 5π/3)

Example 3

Page 40: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

POINTS OF INTERSECTION

These can be found using symmetry or

by noticing that another equation of the circle

is r = -½.

Then, we solve

r = cos 2θ and r = -½.

Example 3

Page 41: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

To find the length of a polar curve r = f(θ),

a ≤ θ ≤ b, we regard θ as a parameter and

write the parametric equations of the curve

as:

x = r cos θ = f(θ)cos θ

y = r sin θ = f (θ)sin θ

Page 42: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

Using the Product Rule and differentiating

with respect to θ, we obtain:

cos sin

sin cos

dx drr

d d

dy drr

d d

Page 43: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

So, using cos2 θ + sin2 θ = 1,we have:2 2

22 2 2

22 2 2

22

cos 2 cos sin sin

sin 2 sin cos cos

dx dy

d d

dr drr r

d d

dr drr r

d d

drr

d

Page 44: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

Assuming that f’ is continuous, we can

use Theorem 6 in Section 10.2 to write

the arc length as:

2 2b

a

dx dyL d

d d

Formula 5

Page 45: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

Therefore, the length of a curve with polar

equation r = f(θ), a ≤ θ ≤ b, is:

22b

a

drL r d

d

Formula 5

Page 46: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

Find the length of the cardioid

r = 1 + sin θ

We sketched it in Example 7 in Section 10.3

Example 4

Page 47: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

Its full length is given by the parameter interval

0 ≤ θ ≤ 2π.

So, Formula 5 gives:2

2 2

0

2 2 2

0

2

0

(1 sin ) cos

2 2sin

drL r d

d

d

d

Page 48: PARAMETRIC EQUATIONS AND POLAR COORDINATES 10. 10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area.

ARC LENGTH

We could evaluate this integral by multiplying

and dividing the integrand by or

we could use a computer algebra system.

In any event, we find that the length of the cardioid is L = 8.

2 2sin