MAT01B1: Calculus with Polar coordinates Dr Craig 23 October 2018
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Today s lecture A recap of polar coordinates and polar curves Tangents to polar curves Symmetry of polar curves Area contained by polar curves Maybe: arc length of polar curves
Polar coordinates: we choose a point O (the origin, or the pole) and define a point using its angle from the polar axis (θ) and its distance from the origin (r).
What does it mean if r < 0? Consider the point ( 2, π 4). We usually think of distances being positive, so where would we expect this point to appear on the plane? Hence ( ( ) 2, 4) π is the same point as 2, 5π 4.
Converting to Cartesian coordinates x = r cos θ y = r sin θ x 2 + y 2 = r 2 tan θ = y x As with complex numbers, make sure that your point is in the correct quadrant.
Example: Sketch the curve r = 1 + sin θ. This type of curve is known as a cardioid.
Limacons The family of curves modelled by the polar equation r = 1 + c sin θ are known as limaçons. (Notice that when c = 1 we have a cardioid.) Example: sketch r = 1 2 sin θ.
Limaçons: r = 1 + c sin θ
Limaçons: r = 1 + c sin θ
Getting to grips with polar curves Exercise 54 in Stewart (Ch 10.3) is a great exercise for thinking logically about the shape of a given polar curve.
Tangents to polar curves: polar curves are defined by r written as some function of θ. That is, r = f(θ). We now want to calculate dy. We proceed as follows: dx dy dx = dy dθ dx dθ To calculate dy dx and we use the fact that dθ dθ x = r cos θ = f(θ) cos θ, y = r sin θ = f(θ) sin θ
Tangents to polar curves dy dx = dy dθ dx dθ = dr sin θ + r cos θ dθ dr cos θ r sin θ dθ Note that if r = 0 (i.e. the point is at the pole) then we have dy dx = tan θ if dr dθ 0
Tangent example: (a) Find the tangent to r = 1 + sin θ at θ = π/3. (b) Find all of the points on the cardioid where the tangent is either vertical or horizontal.
Symmetry If r = f(θ) = f( θ) then the curve is symmetric about the polar axis. If r = f(θ) or f(θ) = f(θ + π) is satisfied then the curve is symmetric about the pole (i.e. curve is unchanged when rotated 180 about the origin). If r = f(θ) = f(θ π) then the curve is symmetric about the line θ = π/2. Symmetry is usually found just by looking at the curve. It can be very useful when applied to area & arc length calculations.
Example: the curve r = cos 2θ is symmetric about the polar axis and about the line θ = π 2.
Area We can use integration to calculate the area of the region bounded by a polar curve. NB: the intuition is different!
Area Area of a circle: A = πr 2. Now think of the formula for the area of a sector of a circle: ( ) θ A = πr 2 = 1 2π 2 r2 θ
Area continued If we split up a region into n slices, where each slice covers an angle of θ, then A i 1 2 (f(θ i )) 2 θ. This gives us an approximation to the area: n 1 A 2 (f(θ i )) 2 θ. i=1 i=1 We then take the limit and define n 1 b lim n 2 (f(θ i )) 2 1 θ = 2 (f(θ))2 dθ a
Area example: find the area enclosed by one loop of the four-leaved rose r = cos 2θ. Answer: 3π/4 1 A = π/4 2 (cos(2θ))2 dθ = π 8 Note: the choice of bounds is not unique 3π/4 5π/4 1 π/4 2 (cos(2θ))2 dθ = Symmetry could also be used: A = 2 π/2 π/4 3π/4 1 2 (cos(2θ))2 dθ = π 8 1 2 (cos(2θ))2 dθ
Area example: find the area inside r = 3 sin θ and outside 1 + sin θ. You must first find the points of intersection. The sketch is extremely important for understanding how to calculate this area. Solution: π.
Next time: arc length of polar curves.