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<cos))) œ0 cos and Cœ< sin ))) œ0 sin . .C .C .) Recall also that œ .B .B .) 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<cos))) œ0 cos and Cœ< sin ))) œ0 sin , we can conclude .C .C .) that œ.B œ .B .) 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.