POLAR CURVES Intersections

Sketch these graphs: r=4 cos θ 𝑟=10 cos 𝜃 𝑟= 1 4 sin 𝜃 r=2 sin 4θ Polar curves KUS objectives BAT Find points of intersection of Polar curves BAT Find Areas bounded by parts of Polar curves Starter: Sketch these graphs: 𝑟=10 cos 𝜃 r=4 cos θ 𝑟= 1 4 sin 𝜃 r=2 sin 4θ 𝑟=6 cos 𝜃 +8 sin 𝜃 r=1+ cos θ 𝑥=𝑟𝑐𝑜𝑠𝜃 𝑦=𝑟𝑠𝑖𝑛𝜃 𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥 𝑟 2 = 𝑥 2 + 𝑦 2

WB16 a) On the same diagram, sketch the curves with equations: 𝑟=3+2 cos 𝜃 and 𝑟=5 −2 cos 𝜃 for −𝜋≤𝜃≤𝜋 b) Find the polar coordinates of the intersection of these curves To find the intersection, we can use the two equations we were given: 3+2𝑐𝑜𝑠𝜃=5−2𝑐𝑜𝑠𝜃 2=4𝑐𝑜𝑠𝜃 0.5=𝑐𝑜𝑠𝜃 𝜃= 𝜋 3 , − 𝜋 3 Using these values of θ, we get r = 2.5 at both points

Using these values of θ, we get WB17 a) On the same diagram, sketch the curves with equations: r = 2 + cosθ and r = 5cosθ b) Find the polar coordinates of the intersection of these curves To find the intersection, we can use the two equations we were given: 2+𝑐𝑜𝑠𝜃=5𝑐𝑜𝑠𝜃 2=4𝑐𝑜𝑠𝜃 π 2 0.5=𝑐𝑜𝑠𝜃 (2.5,π/3) 𝜃= 𝜋 3 , − 𝜋 3 𝒓=𝟐+𝒄𝒐𝒔𝜽 π 0, 2π Using these values of θ, we get r = 2.5 at both points 𝒓=𝟓𝒄𝒐𝒔𝜽 (2.5,-π/3) 3π 2

POLAR CURVES Areas

WB18a To find the area enclosed by the curve, and the half lines θ = α and θ = β, you can use the formula : 𝐴𝑟𝑒𝑎= 1 2 𝛼 𝛽 𝑟 2 𝑑𝜃 0, 2π π 2 π 3π 2 𝒓=𝟏+𝒄𝒐𝒔𝜽 π 6 π 3 Find the area enclosed by the curve, r=1+cosθ, and the half lines θ = π 6 and θ = π 3 , = 1 2 𝜋 6 𝜋 3 1+𝑐𝑜𝑠𝜃 2 𝑑𝜃 notice the 1/2r2θ being familiar as the formula for the area of a sector

WB18b Find the area enclosed by the curve, r=1+cosθ, and the half lines θ = π 6 and θ = π 3 , 1 2 𝜋 6 𝜋 3 1+𝑐𝑜𝑠𝜃 2 𝑑𝜃 = 1 2 𝜋 6 𝜋 3 (1+2 cos 𝜃 + 𝑐𝑜𝑠 2 𝜃 )𝑑𝜃 = 1 2 𝜋 6 𝜋 3 (1+2 cos 𝜃 + 1 2 cos 2𝜃 +1 )𝑑𝜃 = 1 2 𝜋 6 𝜋 3 3 2 +2 cos 𝜃 + 1 2 cos 2𝜃 𝑑𝜃 = 1 2 3 2 𝜃+2 sin 𝜃 + 1 4 sin 2𝜃 𝜋 3 𝜋 6 = 1 2 3𝜋 6 +2 sin 𝜋 3 + 1 4 sin 2 𝜋 3 − 1 2 3 2 𝜋 6 +2 sin 𝜋 6 + 1 4 sin 2 𝜋 6 =1.047

Now we can think about actually Integrating! WB19a Find the area enclosed by the cardioid with equation: r = a(1 + cosθ) 0, 2π π 2 π 3π 2 As the curve has reflective symmetry, we can find the area above the x-axis, then double it… Sketch the graph (you won’t always be asked to do this, but you should do as it helps visualise the question…) So for this question: 𝑟=𝑎(1+𝑐𝑜𝑠𝜃) 𝛼=0 𝛽=𝜋 We will now substitute these into the formula for the area, given earlier: 𝐴𝑟𝑒𝑎= 1 2 𝛼 𝛽 𝑟 2 𝑑𝜃 = 0 𝜋 𝑎 1+𝑐𝑜𝑠𝜃 2 𝑑𝜃 =𝑎 2 0 𝜋 1+2𝑐𝑜𝑠𝜃+ 𝑐𝑜𝑠 2 𝜃 𝑑𝜃 We will need to rewrite the cos2 term so we can integrate it =𝑎 2 0 𝜋 3 2 +2𝑐𝑜𝑠𝜃+ 1 2 𝑐𝑜𝑠2𝜃 𝑑𝜃 = 𝑎 2 0 𝜋 1+2𝑐𝑜𝑠𝜃+ 1 2 𝑐𝑜𝑠2𝜃+ 1 2 𝑑𝜃 Now we can think about actually Integrating!

WB19b Find the area enclosed by the cardioid with equation: r = a(1 + cosθ) =𝑎 2 0 𝜋 3 2 +2𝑐𝑜𝑠𝜃+ 1 2 𝑐𝑜𝑠2𝜃 𝑑𝜃 0 𝜋 3 2 𝜃 + 1 4 𝑠𝑖𝑛2𝜃 = 𝑎 2 + 2𝑠𝑖𝑛𝜃 3 2 𝜋+0+0 − = 𝑎 2 0+0+0 = 3𝜋 𝑎 2 2 Show full workings, even if it takes a while. It is very easy to make mistakes here!

So we would plot r for the following ranges of 4θ WB20a Find the area of one loop of the curve with polar equation : r = asin4θ Think about plotting r = asin4θ From the patterns you have seen, you might recognise that this will have 4 ‘loops’ Sinθ π/2 1 -1 π 2π 3π/2 From the Sine graph, you can see that r will be positive between 0 and π As the graph repeats, r will also be positive between 2π and 3π, 4π and 5π, and 6π and 7π So we would plot r for the following ranges of 4θ 0 ≤ 4θ ≤ π 2π ≤ 4θ ≤ 3π 4π ≤ 4θ ≤ 5π 6π ≤ 4θ ≤ 7π 0 ≤ θ ≤ π/4 π/2 ≤ θ ≤ 3π/4 π ≤ θ ≤ 5π/4 3π/2 ≤ θ ≤ 7π/4 So the values we need to use for one loop are: 3π/4 π/2 π/4 Sometimes it helps to plot the ‘limits’ for positive values of r on your diagram! 𝛽= 𝜋 4 𝑟=𝑎𝑠𝑖𝑛4𝜃 𝛼=0 π 5π/4 3π/2 7π/4

WB20b Find the area of one loop of the curve with polar equation : r = asin4θ 𝐴𝑟𝑒𝑎= 1 2 𝛼 𝛽 𝑟 2 𝑑𝜃 1 2 0 𝜋 4 𝑎𝑠𝑖𝑛4𝜃 2 𝑑𝜃 = 1 2 0 𝜋 4 𝑎 2 𝑠𝑖𝑛 2 4𝜃 𝑑𝜃 1 2 𝑎 2 0 𝜋 4 𝑠𝑖𝑛 2 4𝜃 𝑑𝜃 We will need to write sin24θ so that we can integrate it 1 4 𝑎 2 0 𝜋 4 1−𝑐𝑜𝑠8𝜃 𝑑𝜃 Now this has been set up, we can actually Integrate it! 1 4 𝑎 2 𝜃− 1 8 𝑠𝑖𝑛8𝜃 0 𝜋 4 1 4 𝑎 2 𝜋 4 − 1 8 𝑠𝑖𝑛2𝜋 = 𝜋𝑎 2 16 Important points: You sometimes have to do a lot of rearranging/substitution before you can Integrate Your calculator might not give you exact values, so you need to find them yourself by manipulating the fractions

The region we are finding the area of is highlighted in green WB21a On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves π 2 And b) were done in WB 17 (2.5,π/3) 𝒓=𝟐+𝒄𝒐𝒔𝜽 π 0, 2π The region we are finding the area of is highlighted in green  We can calculate the area of just the top part, and then double it (since the area is symmetrical) 𝒓=𝟓𝒄𝒐𝒔𝜽 (2.5,-π/3) 3π 2

WB20b 𝒓=𝟐+𝒄𝒐𝒔𝜽 𝒓=𝟓𝒄𝒐𝒔𝜽 For the red curve: For the blue curve: 𝑟=2+𝑐𝑜𝑠𝜃 On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves π 2 π 3 𝒓=𝟐+𝒄𝒐𝒔𝜽 π You need to imagine the top part as two separate sections Draw on the ‘limits’, and a line through the intersection, and you can see that this is two different areas The area under the red curve with limits 0 and π/3 The area under the blue curve with limits π/3 and π/2 We need to work both of these out and add them together! 𝒓=𝟓𝒄𝒐𝒔𝜽 3π 2 For the red curve: For the blue curve: 𝑟=2+𝑐𝑜𝑠𝜃 𝑟=5𝑐𝑜𝑠𝜃 𝛼=0 𝛼= 𝜋 3 𝛽= 𝜋 3 𝛽= 𝜋 2

WB21c For the red curve: For the blue curve: 𝑟=2+𝑐𝑜𝑠𝜃 𝑟=5𝑐𝑜𝑠𝜃 𝛽= 𝜋 3 On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves For the red curve: For the blue curve: 𝑟=2+𝑐𝑜𝑠𝜃 𝑟=5𝑐𝑜𝑠𝜃 𝛽= 𝜋 3 𝛼= 𝜋 3 𝛽= 𝜋 2 𝛼=0 1 2 𝛼 𝛽 𝑟 2 𝑑𝜃 For the red curve: Sub in the values from above  Also, remove the ‘1/2’ since we will be doubling our answer anyway! 0 𝜋 3 2+𝑐𝑜𝑠𝜃 2 𝑑𝜃 Square the bracket 0 𝜋 3 4+4𝑐𝑜𝑠𝜃+ 𝑐𝑜𝑠 2 𝜃 𝑑𝜃 Replace the cos2θ term with an equivalent expression (using the equation for cos2θ above) 0 𝜋 3 4+4𝑐𝑜𝑠𝜃+ 1 2 𝑐𝑜𝑠2𝜃+ 1 2 𝑑𝜃 Group like terms, and then we can integrate! 0 𝜋 3 9 2 +4𝑐𝑜𝑠𝜃+ 1 2 𝑐𝑜𝑠2𝜃 𝑑𝜃

WB21d For the red curve: For the blue curve: 𝑟=2+𝑐𝑜𝑠𝜃 𝑟=5𝑐𝑜𝑠𝜃 𝛽= 𝜋 3 On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves For the red curve: For the blue curve: 𝑟=2+𝑐𝑜𝑠𝜃 𝑟=5𝑐𝑜𝑠𝜃 𝛽= 𝜋 3 𝛼= 𝜋 3 𝛽= 𝜋 2 𝛼=0 0 𝜋 3 9 2 +4𝑐𝑜𝑠𝜃+ 1 2 𝑐𝑜𝑠2𝜃 𝑑𝜃 For the red curve: Integrate each term, using ‘standard patterns’ where needed… 9 2 𝜃+4𝑠𝑖𝑛𝜃+ 1 4 𝑠𝑖𝑛2𝜃 0 𝜋 3 Sub in the limits separately (as subbing in 0 will give 0 overall here, we can just ignore it!) 9 2 𝜋 3 +4𝑠𝑖𝑛 𝜋 3 + 1 4 𝑠𝑖𝑛 2𝜋 3 Calculate each part (your calculator may give you a decimal answer if you type the whole sum in) 3𝜋 2 +2 3 + 3 8 Write with a common denominator 12𝜋 8 + 16 3 8 + 3 8 Group up 12𝜋+17 3 8

WB21e For the red curve: For the blue curve: 𝑟=5𝑐𝑜𝑠𝜃 𝛼= 𝜋 3 𝛽= 𝜋 2 On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves For the red curve: For the blue curve: 𝑟=5𝑐𝑜𝑠𝜃 𝐴𝑟𝑒𝑎= 12𝜋+17 3 8 𝛼= 𝜋 3 𝛽= 𝜋 2 1 2 𝛼 𝛽 𝑟 2 𝑑𝜃 For the blue curve: Sub in the values from above  Also, remove the ‘1/2’ since we will be doubling our answer anyway! 𝜋 3 𝜋 2 5𝑐𝑜𝑠𝜃 2 𝑑𝜃 Square the bracket 𝜋 3 𝜋 2 25 𝑐𝑜𝑠 2 𝜃 𝑑𝜃 Replace the cos2θ term with an equivalent expression (using the equation for cos 2θ above) 𝜋 3 𝜋 2 25 1 2 𝑐𝑜𝑠2𝜃+ 1 2 𝑑𝜃 We can move the ‘1/2’ and the 25 outside to make the integration a little easier 25 2 𝜋 3 𝜋 2 𝑐𝑜𝑠2𝜃+1 𝑑𝜃

WB21f For the red curve: For the blue curve: 𝑟=5𝑐𝑜𝑠𝜃 𝛼= 𝜋 3 𝛽= 𝜋 2 On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves For the red curve: For the blue curve: 𝑟=5𝑐𝑜𝑠𝜃 𝐴𝑟𝑒𝑎= 12𝜋+17 3 8 𝛼= 𝜋 3 𝛽= 𝜋 2 25 2 𝜋 3 𝜋 2 𝑐𝑜𝑠2𝜃+1 𝑑𝜃 For the blue curve: Integrate each term, using ‘standard patterns’ if needed… 25 2 1 2 𝑠𝑖𝑛2𝜃+𝜃 𝜋 3 𝜋 2 Sub in the limits (we do need to include both this time as neither will cancel a whole section out!) 25 2 1 2 𝑠𝑖𝑛𝜋+ 𝜋 2 − 1 2 𝑠𝑖𝑛 2𝜋 3 + 𝜋 3 Calculate each part as an exact value 25 2 𝜋 2 − 3 4 + 𝜋 3 Write with common denominators 25 2 6𝜋 12 − 3 3 12 + 4𝜋 12 Group up and multiply by 25/2 50𝜋−75 3 24

Add these two areas together to get the total area! WB21g On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθ Find the polar coordinates of the intersection of these curves Find the exact value of the finite region bounded by the 2 curves For the red curve: For the blue curve: 𝐴𝑟𝑒𝑎= 12𝜋+17 3 8 𝐴𝑟𝑒𝑎= 50𝜋−75 3 24 Add these two areas together to get the total area! 12𝜋+17 3 8 + 50𝜋−75 3 24 Write with a common denominator 36𝜋+51 3 24 + 50𝜋−75 3 24 Add the numerators 86𝜋−24 3 24 Divide all by 2 43𝜋−12 3 12 These questions are often worth a lot of marks! Your calculate might not give you exact values for long sums, so you will need to be able to deal with the surds and fractions yourself!

One thing to improve is – KUS objectives BAT Find Areas bounded by parts of Polar curves BAT Find points of intersection of Polar curves self-assess One thing learned is – One thing to improve is –

Practice Ex 7D

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