Polar Functions - Area

BC CALCULUS

5047 Polar Area BC Bonus

AP Calculus

REVIEW Polar Curves:

cos( )sin( )

r cr ar a

cos( )sin( )

r b ar b a

Limaζon

Circles

b < a with loop

b = a Cardioid

b > a no loop

cos( )sin( )

r a nr a n

Rose n(odd) = n petals

n(even) = 2n petals

a gives the Diameter (or length of a Leaf)

REVIEW Intersections:Set the Equations equal and solve.Check the pole independently in each curve.

4sin( )4cos( )

rr

REVIEW Intersections:Set the Equations equal and solve.Check the pole independently in each curve.

32 2cos( )

rr

REVIEW Intersections:Set the Equations equal and solve.Check the pole independently in each curve.

2 (2 )2sin( )

r cosr

Lemma: Trig substitution

2 2

2 2

2 2

(2 ) cos ( ) sin ( )

1 1cos(2 ) 2cos ( ) 1 cos ( ) cos(2 )2 2

1 1cos(2 ) 1 2sin ( ) sin ( ) cos(2 )2 2

cos x x x

x x x x

x x x x

Area of a Circular Sector

REM: Calculus works with RADIANS.

2

2

2

2

( )2

12

r

r

r

Ar

rA

A r

Area of a Polar Region

212

rA r

( ) 0( )

lim ( )

b xh f xA f x x

TA f x dx

REM:

Illustration:Find the area of the region bounded by the cardioid.

2 2cos( )r

Example:Find the area of the region bounded by the one leaf of the rose.

4sin(3 )r

Example: Region of IntersectionFind the area of the region in the intersection between the curves.

sin( ) cos( )r and r

Example: Region between curvesFind the area of the region inside the cardioid and outside the circle.

2 2cos( )3

rr

LAST UPDATE

• 03/19/12

• Assignment : p. 558 # 43 – 59 odd

Area of a Circular SectorREM: Calculus works with RADIANS. -

360 2 2

2 2

o r

o

r

r

r

r

r

2 2

2 2

r

Ar r

r rrA

212

rA r