MATH 37 Lecture Guide UNIT 4

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ctureGuideUNIT4albabierra

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UNIT 4. POLAR COORDINATES and POLAR GRAPHS MATH 37 LECTURE GUIDE

Objectives: By the end of the unit, a student must correctly and confidently be able to: convert Cartesian coordinates to polar coordinates, and vice-versa; convert Cartesian equations to polar equations , and vice-versa; graph polar curves;

determine intersections of polar curves; and find area of polar regions.

It will be helpful to review your circular function values.

10 cos 2

3

6

cos

2

2

4cos

2

1

3

cos 0

2cos . . .

00 sin 2

1

6

sin

2

2

4sin

2

3

3

sin 1

2sin . . .

__________________________

4.1 Relation between Cartesian and Polar Coordinate Systems (TC7 pp. 790-796)

Cartesian coordinate: y,xP x: directed distance from the y axis

y : directed distance from the x axis

Polar coordinate: ,rP r r : distance from the origin (maybe less than 0) : angle in standard position made by segment

OP with the positive x axis(measured in radians)

Remarks:

1. The polar coordinate of a point is NOT unique.

2. A negative r is a distance towards the oppositedirection of .

2

axis

POLE

y,xy

x

TO DO!!! Plot the following points in polarcoordinates.

1.

62

,

2.

4

53

,

3.

3

21

,

4.

3

25

,

5.

6

54

, ,

6

114

, ,

6

74

,

and

64

, are coordinates

of the same point.

polar

axis


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TRY THIS!!!

Determine the other coordinates of the point

6

53

,P such that:

a.) 00 ,r b.) 00 ,r c.) 00 ,r

MUST REMEMBER!!! CONVERSIONS

Cartesian coordinate: y,xP

to POLAR: 22 yxr

For , if 0x , the point is either at the2

- axis (if 0y )

or23 - axis (if 0y ).

If 0y , the point is either at the polar axis (if 0x )

or - axis (if 0x ).If y,xP is in the 1st or 4th quadrant,

x

ytanArc .

If y,xP is in the 2nd or 3rd quadrant,

x

ytanArc .

Polar coordinate: ,rP

to CARTESIAN: cosrx sinry

TO DO!!!

Convert the following to polar coordinates. Convert the following to Cartesian coordinates.

1. 44 , 5.

6

73

,

2. 43 , 6.

3

22

,

3. 13 , 7.

4

35

,

4. 40

,


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Converting Cartesian equations to polar, and vice-versa

Convert the equation of a circle 422 yx to polar.

Remark: The polar equation orr , where or is a constant, is a graph of a circle centered at the

pole of radius or .

The following are circles tangent to the pole where a is the diameter of the circle:

cosar cosar sinar sinar The following are equation of lines:

vertical line: acosr

hoizontal line: bsinr diagonal line through the pole: o

TO DO!!! Sketch the graphs of the following.

1. cosr 8 2. sinr 10

TO DO!!!

1. Convert the Cartesian equation 2xy to its polar form. Simplify the expression.


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4.2 Graphs of Polar Equations (TC7 pp. 798-803)

Consider the polar equation cosr 22 .

0 6

4

3

2

3

2

4

3

6

5

r 0 32 22 1 2 3 22 32 4

6

7

4

5

3

4

2

3

3

5

4

7

6

11 2

r 32 22 3 2 1 22 32 0

Sketch the graph of cosr 22 using the data from the given table.

Use the following approximate values.

7313 . 4112 .

MUST REMEMBER!!! LIMAONS

The graphs of cosbar cosbar sinbar sinbar 00 b,a are called limaons. The values of a and b determine the limaons shape.

1b

a 1

b

a 21

b

a 2

b

a


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To graph limaons: sinbar

1. Identify the type of limaon by consideringb

a.

2. Plot points at 2

3

20

,,, . cosbar cosbar 3. Using the four points you plotted, trace the

proper graph as identified. Remember thatlimaons are smooth, rounded graphs.

The illustration on the right should give you somehints as to what axis your limaon should be

pointing. sinbar


1. sinr 21

2. cosr 23


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MUST REMEMBER!!! ROSES

The graphs of nsinar ncosar where n is a positive integerare called roses. If n is even, there are n2 petals/leaves.

If n is odd, there are n petals/leaves.

To graph roses:

1. Determine the number of petals/leaves.

2. For the tips of the petals, solve for such that 1nsin or 1ncos .3. The curve passes through the pole (origin) at such that 0nsin or 0ncos .

4. Using the points you solved for, trace the proper graph as identified. Remember that petals ofa rose are smooth and rounded.


3. cosr 3

4. sinr 33


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Examples.

A five-leaf rose An 8-leaf rose


1. 23sinr 0r at

max r at

min r at

2. 34cosr 0r at

max r at

min r at


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LEMNISCATES(propellers): 222 cosar 222 sinar

For 222 cosar , 2r is defined only when 02 cos

2

4

7

4

5

4

3

40 ,,, .

For 222sinar

, 2

ris defined only when 02

sin

2

3

2

0

,, .

Examples.

For 222 cosar , 2r is defined only when 02 cos

4

7

4

5

4

3

4

,, .

For 222 sinar , 2r is defined only when 02 sin

2

2

3

2,, .

Examples.

SPIRALS: 0 ,r (Spiral of Archimedes) 1 ,lnr (Logarithmic Spiral)Examples.


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Supplement. Test of Symmetry.

A polar graphs is

i. symmetric with respect to thepolar axis if an equivalent equation is obtained when ,r isreplaced by either ,r or ,r ;

ii. symmetric with respect to the 2

axis if an equivalent equation is obtained when ,r is

replaced by either ,r or ,r ; oriii. symmetric with respect to thepole if an equivalent equation is obtained when ,r is

replaced by either ,r or ,r .

Exercise Set. Try to solve the following.

Sketch the graphs of the curves defined by the given polar equations. Identify the graphs. Use themethods prescribed to graph each type of polar curves.

1. 04

3

r 9. 57 cosr

2. cosr 12

10. 45 sinr 3. sinr 25 11. 2162 cosr 4. cosr 33 12. 252 sinr 5. cosr 42 13. 01 ,r (spiral)6. sinr 55 14. 0 ,er (spiral)

7. sinr 31 15. 25

cosr CHALLENGE!!!

8. cosr 26 16. 332 sinr CHALLENGE!!!

17. Investigate the family of curves defined by the polar equations ncosr , where n is a positiveinteger. How do the number of leaves depend on n ? Do the same for nsinr .

18. Find the conditions on k to determine how the spiral kr unwinds (whether clockwise orcounter-clockwise).

19. In many cases, polar graphs are related to each other by rotation.

a. How are the graphs of

3

1sinr and

3

1sinr related to the graph of

sinr 1 ?b. In general, how is the graph of kfr related to the graph of fr ?

20. Investigate the family of curves given by ncosbar , where a , b and are realnumbers and n is a positive integer.

a. How are the graphs when 0 and 0 related?b. How does the graphs change as n increases?

_________________________


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4.3 Intersections of Polar Curves (TC7 pp. 810-12/TCWAG pp. 622-624)

Equations of polar curves are NOT unique. The polar curve defined by the equation fr

is the same as the curve given by nfrn 1 , where n is an integer. This is because of the

periodic nature of circular functions.


Determine the area of the following region. Try using both vertical and horizontal strips, if possible.

1. the region bounded by xsiny and xcosy ,22

x .

To solve for intersections of polar curves, the different equations identifying the curves shouldbe considered. In some cases, curves intersect at a point but at different values of . For example,the pole is given by ,0 , for any as long as 0r .

TO DO!!!

1. Identify the other equations which give the curve defined by 22 sinr .

2. Identify the other equations which give the curve defined by cosr 1 .

MUST REMEMBER!!!

Solving intersections of polar curves given by fr:C 1 and gr:C 2 .

1. Determine all the distinct equations of the curves using nfrn 1 and

ngrn 1 .

2. Solve every pair of equations defining the two curves. This should exhaust every possible

pair of equation for 1C and for 2C .

3. For the intersection at the pole, set both equations to zero ( 0f and 0g ) and solve

for . Note that the two curves can intersect at the pole but at different values of .


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Solve for the intersections of the following pairs of graphs. It will help to identify first if the equivalentequations are needed or not. Graphs can greatly help determine the number of intersections.

1. cosr 1 and sinr 1 6. cosr 1 and sinr 1 2. 6r and cosr 44 7. 23 sinr and 23 cosr 3. cosr 33 and sinr 3 8. 222 cosr and 1r 4. cosr 1 and sinr 1 9. 22 sinr and sinr 2 5. 5r and cosr 21

5

10. 242 cosr and sinr 22

_________________________

TO DO!!!

1. Solve for the intersections of the curves given by 221 sinr:C and 12 r:C .

2. Solve for the intersections of the curves given by cosr:C 411 and 22 r:C .


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4.4 Area in Polar Coordinates (TC7 pp. 807-812)

Area of a sector of a circle = 22

1 r , where is in radians

Consider the region bounded by 0 fr from to .Dividing the sector to n sub-sectors,

nn... 1210 such that the i th sector is from 1 i to i .

The area of the i th sector is

iii rA 22

1 iif 2

2

1 .

The area of the region is given by

n

i

ii

n

i

i fAA

1

2

1

2

1 .

Hence,

n

i

iin

flimA

1

2

2

1 .

Thus, area of the region bounded by the curve fr , , is given by

df 22

1.

Suppose 0 gf for .The area of the region bounded by the curves

fr and gr , ,

is given by

dgf 222

1.


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TO DO!!! SET-UP the definite integral that will solve for the volume of the following regions.

1. the region inside the cardioid cosr 55

2. the shaded region bounded by r

3. the region inside the circle 3r

and the cardioid cosr 22

4. the region outside the circle 3r

but inside the cardioid cosr 22

5. the region inside the circle 3r

but outside the cardioid cosr 22

So, what is the total area of the region enclosed by 3r and cosr 22 ?


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Set-up the definite integrals that will solve for the area of the following regions. Then, if possible or theintegral is easy to evaluate, determine the area of the region.

1. the region enclosed by one petal of the rose 34 cosr 2. the region enclosed by cosr 25 3. the region inside the small loop of sinr 42 4. the region inside the large loop of sinr 42 5. the intersection of the regions enclosed by 22 sinr and sinr 2 6. the intersection of the regions enclosed by 23 sinr and 23 cosr 7. the region inside cosr 33 but outside sinr 33 8. the region outside 2r but inside 282 cosr 9. the region inside the loop of

cosr

41but outside

1r

10. the region outside cosr 1 but inside sinr 3

TO DO!!! SET-UP the definite integral that will solve for the volume of the following regions.

5. the region inside the rose 22 sinr but outside the circle 1r

6. the region inside the circle 3r but outside

the limaon with a loop cosr 21

How will you solve for the region inside the loop of cosr 21 ?


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11. Determine the value of a for which the area of the region enclosed by the cardioid cosaar is 9 square units.

12. Find the area of the region inside the cardioid cosaar but outside the circle cosar 2 .

13. Find the area of the region inside the circles given by sinar 2 and cosbr 2 with 0b,a .

14. Consider the two shaded regions below. The one on the left is given by the curve 0 ,r .The one on the right is given by the curve 1 ,lnr . Determine the values of at which thecurves intersect the axes up to the second revolution of the curve. The, set-up the integrals that

will solve for the area of the shaded region.

15. A classmate computes the area of the region in #12. The answer he/she gave was

2

0

22221

2

1

dcoscosa . Is this answer correct? Justify your answer.

16. A goat is tethered to the edge of a circular

pond of radius a by a rope of length ka

( 20 k ). Using polar coordinates,

determine the grazing area. In the figure on

the right, the grazing area is the shaded

region.

17. CHALLENGE!!!Find the area of the loop of cossecr 2 . The graph of this equation iscalled a strophoid.

18. CHALLENGE!!!Find the total area inside the petals of the rose ncosar where is a positiveinteger.

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END OF UNIT 4 Lecture Guide

MATH 37 Lecture Guide UNIT 4

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