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

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 , 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

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  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

Exam Tip

  • 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

Worked Example

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

7-1-2-sp1-polcurv-we-qu

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.

7-1-2-edex-alevel-fm-polar-we1-soltn-i

7-1-2-edex-alevel-fm-polar-we1-soltn-ii

 

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

  • 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

7-1-2-edx-a-fm-fig2-polarcurvearea

  • 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

Exam Tip

  • The use of symmetry in these problems can make them a lot easier so do always look to use it
  • Calculators may be able to evaluate integrals but remember they usually expect x to be the ‘input’ variable
    • Calculators may not always produce exact values so check what is required by the question

Worked Example

Find the exact area of one loop of the curve with polar equation .

7-1-2-edex-alevel-fm-polar-we2-soltn

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

7-1-2-edexcel-al-fm-polar-multiple-areas

Exam Tip

  • Graph sketches do not have to be accurate, but should enable you to visualise the problem and get an idea of where intersections and half-lines are
  • Look out for when exact areas are required and whether your calculator can produce these using its integration function

Worked Example

A sketch of the polar curves defined by the following equations is shown below

7-1-2-sp3-shadpol-we-qu

a) Find the area labelled .

7-1-2-edex-alevel-fm-polar-we3-soltn-a

a) Find the area labelled .

7-1-2-edex-alevel-fm-polar-we3-soltn-b

 

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