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Edexcel A Level Further Maths: Core Pure:復習筆記7.1.2 Calculus with Polar Coordinates

Category: A-level課程, 國際課程, 教材筆記, 福利干貨 Date: 2022年10月20日下午4:13

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