Section 12.6 – Area and Arclength in Polar Coordinates

12.2

Integrals in Polar Coordinates

12.2

Area of a Polar Curve:2

1

21r d

2

2

2

2

2

cos2 2cos 1

1cos 1 cos2

21

cos 2 1 cos421

cos 3 1 cos62

2

2

2

2

cos2 1 2sin

1sin 1 cos2

21

sin 2 1 cos421

sin 3 1 cos62

A graphic of the area inside one loop of the curve r = 2 Sin(3t)

http://clem.mscd.edu/~talmanl/MOOVs/SimplePolarArea/SimplePolarArea.MOV

A graphic of the area inside one loop of the curve r = 2 Sin(3t) but outside of the circle r = 1

Calculus@Internet

http://clem.mscd.edu/~talmanl/MOOVs/HardPolarAreaI/HardPolarAreaI.MOV

A graphic of the area inside the curve r = 1 + Cos(2t) but outside the curve r = Cos(t).

The total area of the region enclosed

by the polar graph of is r 1 sin

2

2

0

11 sin d

2

A)2

B)

3C)

2D) 2

5E)

2

2

2

0

11 2sin sin d

2

2

0

1 1 cos21 2sin d

2 2

20

1 3 12cos sin2 |

2 2 4

3 31 0 0 1 0 C

2 2

NO CALCULATOR

NO CALCULATOR

2

0

0

2

0

0

2

0

The area of the closed region bounded by the polar graph of

r 1 cos is given by:

A. 1 cos d

B. 1 cos d

C. 2 1 cos d

D. 1 cos d

E. 2 1 cos d

2

2

0

0

1 cos d

1r d

2

12 1 d

D

cos2

NO CALCULATOR: Find the area of the region inside thecircle r = 4 and outside r 4 cos

/ 2

2 2

/ 2

14 4 cos d

2

/ 2

2

/ 2

116 16 8cos cos d

2

/ 2

2

/ 2

18cos cos d

2

/ 2

/ 2

1 1 cos28cos d

2 2 2

/ 2/ 2

1 14sin sin2 |

4 8

4 0 4 0 88 8 4

The area of the region enclosed by the graph of the polar

curve isr 2 2cos

A) 4.712B) 9.424C) 18.849D) 37.699E) 75.398

2

2

0

12 2cos d 18.849 C

2

CALCULATOR REQUIRED

The approximate total area of the region enclosed by the

polar graph of is: r sin2

A) 0.393B) 0.785C) 1.178D) 1.571E) 1.873

/ 2

2

0

14 sin2 d 1.571 D

2

CALCULATOR REQUIRED

Set up to definite integral to find the area inside the smaller loop of r 1 2cos

0, 2

2 4,

3 3

4 / 3

2

2 / 3

11 2cos d

2

CALCULATOR REQUIRED

Find the area inside 2 2r 2a sin3

0,2

/ 3

2

0

16 2a sin3 d

2

2 / 30

2 2

2

6a cos3 |

3

2a 2a

4a

CALCULATOR REQUIRED

Length of an Arc in Polar Coordinates

x f cosx r cos

y r sin y f sin

2 2dx dy

L dd d

dxf ' cos f sin

ddy

f ' sin f cosd

2 2L f ' f d

Find the arc length from 0 to 2 for the cardoid r 4 4cos .

CALCULATOR REQUIRED

r 4 4cos

r ' 4sin

2

2 2

0

4 4cos 4sin d

32

Find the arc length from 0 to for the cardoid r sec .3

NO CALCULATOR

r sec

r ' sec tan

/ 3

2 2

0

sec tan sec d

/ 3

2 2

0

sec tan 1 d

/ 3

2

0

sec d

/ 3

0tan |

3