Polar Coordinates and Calculus

Recall that in the Polar coordinate system the coordinates ) represent
<ß  the directed distance from the pole to the point and the directed angle, counterclockwise from the polar axis to the segment from the pole to the point. A polar function would be of the form: ) . < œ 0
 To find the slope of the tangent line of a polar graph we will parameterize the equation. ) ( Assume is a differentiable function of ) ) <œ0
 0 ¾Bœ

1Þ Find the derivative of < œ # sin )

Remember that horizontal tangent lines occur when and that vertical tangent lines occur when

page 1 2 Find the HTL and VTL for cos ) and sketch the graph. Þ < œ #
" 

page 2 Recall our formula for the derivative of a function in polar coordinates. Since Bœ

Solutions obtained by setting < œ ! gives equations of tangent lines through the pole.

3Þ Find the equations of the tangent line(s) through the pole if <œ#sin #Þ)

page 3 To find the arc length of a polar curve, you have two options.

1) You can use the parametrization of the polar curve:

cos))) cos and sin ))) sin Bœ< œ0
 Cœ< œ0
 then use the arc length formula for parametric curves:

" w # w # 'α ÈÐBÐ) ÑÑ ÐCÐ ) ÑÑ . ) or,

2) You can use an alternative formula for arc length in polar form:

" <# Ð.< Ñ # . ) 'αÉ . )

(Sometimes this gives a cleaner integral to solve.)

# ) 4. Find the arc length of the curve <œ%-9= Ð# Ñ

page 4 To find the points of intersection of two polar curves ) and ) <œ0
 <œ1
 we will solve a system of equationsÞ Because there is not a unique representation for each polar coordinate we sometimes need to look for alternate forms of each polar equation. An alternate form for a polar equation ) is given by the equation 8 ) 1 For <œ0


 "<œ08Þ example an alternate form for the equation < œ # sin ) is sin ) 1 which is equivalent to sin ) <œ#
  <œ # Þ 5Þ Find the points of intersection of <œ#sin #) and <œ"Þ

page 5 The development of the formula for the area of a polar region is similar to the development of the area of a rectangular region except rather than summing the areas of rectangles we sum the areas of sectors of a circle.

Theorem: If is continuous and nonnegative on the interval α " then the 0 cß d ß area of the region bounded by the graph of ) and the radial lines < œ 0
 " " )α)"œ and œ is given by <.# ) #(α

page 6 Find the following areas

6Þ The area enclosed by <œ# sin ) Þ

7Þ The area enclosed by <œ#cos $ )

page 7 8Þ The area enclosed by one loop of <œ## sin # )

page 8 9Þ The area of the inner loop of < œ " # cos )

page 9 10Þ The area outside <œ# and inside <œ% cos )

page 10 11Þ The area common to <œ##sin ) and <œ"

page 11 12Þ Find the area common to <œ%sin) and <œ% cos ) Þ

page 12