Calculus with Polar Coordinates (Edexcel A Level Further Maths): Revision Note

Finding Tangents to Polar Curves

What is the gradient/tangent of a polar curve?

• Gradients (and tangents) are the same as using Cartesian coordinates

  • i.e. a gradient of 1 in Cartesian coordinates is still a gradient of 1 in polar coordinates

  • a 45° line from “bottom left” to “top right” is a gradient of 1 in both systems

  • the equation of a tangent to a polar curve should be written in polar form

How do I find the tangents to a polar curve?

• Finding the gradient - and so the equation of a tangent - to a polar curve is based on parametric differentiation in Cartesian form

• Since ,  and , it follows that

  • 

  • 

• Then, using parametirc differentiation the gradient is given by

  • 

  • From which the Cartesian equation can be found

  • Which can then be converted into polar form using  and 

How do I find horizontal and vertical tangents to a polar curve?

• Many questions only concern tangents that are horizontal and/or vertical to the curve

• Horizontal tangents are described as being “parallel to the initial line”

  • Horizontal tangents occur where 

• Vertical tangents are described as being “perpendicular to the initial line”

  • Vertical tangents occur where 

• Questions require finding the coordinates of points that have horizontal or vertical tangents (rather than finding the equations of the tangents)

  • Coordinates should be in polar form, i.e.

• In some cases, both  and  at a particular point

  • Under these cases the polar curve has a cusp

  • But vice versa is not necessarily true

    • A polar curve with a cusp does not necessarily mean  and 

Finding Areas enclosed by Polar Curves

To find the area enclosed by a polar curve (or part of) it is first crucial to know how to find the area of a sector in polar coordinates

How do I find the area of a sector given by a polar curve?

• In polar coordinates, the area of a sector, A,  is given by

where and

• The sector is bounded by the curve  and the two half-lines  and 

  • This is given in the formula booklet

• If  is constant then the formula gives the area of the sector of a circle with centre angle 

What is meant by the area enclosed by a polar curve?

• The area enclosed by a polar curve refers to an area bounded by a curve  between the half-lines and 

• This can be considered as the area created by a 'sweeping' hand of a clock (but going anticlockwise!) moving between  and 

  • The integral calculates the sum of an infinite number of sectors which start at  and end at 

  • This is the polar equivalent of the sum of an infinite number of rectangles under a curve in Cartesian coordinates

How do I find the area enclosed by a polar curve?

• STEP 1
  If not given, a sketch of the curve is helpful
  Identify the half-lines  and  between which the area lies
  This may involve solving equations
  Always look for symmetry – many problems can be found by finding “half the area” and “doubling” – for example only finding an area above the initial line

• STEP 2
  Find  and manipulate it into an integrable form This may involve using trigonometric identities and/or common integration techniques such as reverse chain rule,  'adjust and compensate'
  Set up the integral using the formula 

• STEP 3
  Evaluate the integral and interpret the answer Remember to double/scale-up the integral value to find the area if symmetry has been used

Finding Areas enclosed by Multiple Polar Curves

What is meant by the area enclosed by multiple polar curves?

• An area enclosed by multiple polar curves could be

  • an area between two polar curves

  • an area partially enclosed by one polar curve and partially enclosed by another

How do I find the area enclosed by multiple polar curves?

• STEP 1
  If not given, a sketch, on the same diagram, of the curves is helpful
  Identify any half-lines that are needed by looking for intersections between the curves
  Identify any relevant values of θ such that r=0 (i.e. intersections with the pole)
  This may involve solving equations in relevant ranges of θ
  Look for symmetry to simplify the problem

• STEP 2
  Find  for both curves, manipulating them into integrable forms

  This may involve using trigonometric identities and or common integration techniques such as reverse chain rule, ‘adjust and compensate’
  Set up an integral for each partial area using the formula
   

• STEP 3
  Evaluate the integrals
  Double/scale-up each integral as necessary if symmetry has been used
  Total the partial integrals to find the entire area required