Workbook: Polar Curves Intersections, Areas and Tangents

WB16 a) On the same diagram, sketch the curves with equations:r=3+2 cosθ and r=5−2cosθ for −π ≤θ≤π

b) Find the polar coordinates of the intersection of these curves

WB17 a) On the same diagram, sketch the curves with equations:r = 2 + cosθ and r = 5cosθb) Find the polar coordinates of the intersection of these curves

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Workbook: Polar Curves Intersections, Areas and Tangents

To find the area enclosed by the curve, and the half lines θ = α and θ = β, you can use the formula :

area=12∫αβ

r2dθ

WB18 Find the area enclosed by the curve, r=1+cosθ, and the half lines θ = π6 and θ =

π3 ,

WB19 Find the area enclosed by the cardioid with equation: r = a(1 + cosθ)

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0, 2π

π 2

π

3π 2

π 6

π 3

Workbook: Polar Curves Intersections, Areas and Tangents

WB20 Find the area of one loop of the curve with polar equation : r = asin4θ

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Workbook: Polar Curves Intersections, Areas and Tangents

WB21 a) On the same diagram, sketch the curves with equations: r = 2 + cosθ r = 5cosθb) Find the polar coordinates of the intersection of these curvesc) Find the exact value of the finite region bounded by the 2 curves

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Workbook: Polar Curves Intersections, Areas and Tangents

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Workbook: Polar Curves Intersections, Areas and Tangents

The equation y = rsinθ represents changes in the vertical direction. When dy/dθ is 0, that means that there is no movement in the vertical direction (the change in y with respect to a change in θ is 0)

When dx/dθ is 0, that means that there is no movement in the horizontal direction (the change in x with respect to a change in θ is 0). Therefore, if dy/dθ is 0, the curve is perpendicular to the ‘initial line’

WB22 Find the coordinates of the points on: r = a(1 + cosθ)Where the tangents are parallel to the initial line θ = 0

WB23 Find the coordinates and the equations of the tangents to the curve :r=asin2θ, 0 ≤ θ ≤ π/2 Where the tangents are:

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Workbook: Polar Curves Intersections, Areas and Tangents a) Parallel to the initial line Give answers to 3 s.f where appropriate

WB24b) Perpendicular to the initial line, Give answers to 3 s.f where appropriate

WB25 Prove that for: r=( p+qcos θ), p and q both > 0 and p ≥ q

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Workbook: Polar Curves Intersections, Areas and Tangents

SUMMARY

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Workbook: Polar Curves Intersections, Areas and Tangents

In polar coordinates, a point is given by

What is r ?

What is ?

When sketching polar coordinates we draw a table of

values. What does it mean for the curve

when

The value of r comes out negative?

The equation of the curve is a function

of only?

The equation of the curve is a function

of only?

When converting from polar coordinates to Cartesian coordinates we use the

relationships

When converting from Cartesian coordinates to polar coordinates we use

the relationships

The formula to calculate the area bounded by a polar curve is

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