Edexcel A Level Further Maths: Core Pure:复习笔记7.1.2 Calculus with Polar Coordinates

• If not provided, sketch the graph of the polar curve
  • This will help you to spot how many horizontal/vertical tangents there are
  • You could use a graphical calculator to help you do this

A sketch of the polar curve , with equation , where  is shown below.

Find the coordinates of the points on  where the tangents are (i) parallel, (ii) perpendicular to the initial line, giving values to 2 significant figures where appropriate.

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

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

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

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