Areas and lengths in polar coordinates By parametric equation - PowerPoint PPT Presentation

Different ways of representing curves on the plane As the set of points satisfying an equation � Given as a function � {(x,y) : G(x,y)=0} � {(x,y), y =f(x)} � Example {(x,y) : x 2 + y 2 =25} Example {(x,y) : y =x 2 +x-0.5 Areas and lengths in polar coordinates By parametric equation � x=f(t), y=g(t), t in [a,b] � Example x=sin(t), y=sin(2t), t in [0,2 π ] A different way of Fix � representing a point on the a point (the pole) and � plane: polar coordinates (4, π /6 ) a ray from the point (the polar axis) Coordinate systems are used to locate the position of a point. (2 √ 3,2) � A point P has polar coordinates (r, ϴ ) if � 1. The distance form the pole is |r|. � 2. ϴ is the measure of the directed angle starting at the polar axis and ending at P = ( r , θ ) the ray from the pole to P. r = directed distance In rectangular or cartesian coordinates : � The plane is “organized” as grid of In polar coordinates : � We break up the plane with circles horizontal and vertical line lines. � O θ = directed angle Polar centered at the origin and with rays A point is labeled with a pair of Pole (Origin) axis emanating from the origin. � numbers that correspond to the We locate a point as the intersection of vertical and the horizontal line the a circle and a ray. point belongs.

How to plot a point P with polar coordinates (r, ϴ ) How to plot a point P with polar coordinates (r, ϴ ) ❖ If r=0, P is the pole. � Plot the points with polar coordinates ❖ If r>0, rotate the polar axis an given below. � angle ϴ (counterclockwise if ❖ (0, 27) � ϴ >0, clockwise otherwise) ❖ (4, π /6) � and place P on this ray at distance r from the pole. � ❖ (-3, π /2) � Points can be represented in ❖ If r<0, proceed as if r>0, but more than one way in polar place P the point in the coordinates. � opposite ray, at distance -r Find more ways to represent the from the pole. points above. r= (x 2 +y 2 ) 1/2 � Example ϴ = arctan(y/x) The polar coordinates If (x,y) in rectangular of the point are (r, ϴ ) coordinates is given x=r.cos ϴ � 1. Find polar coordinates for a the point with rectangular y=r.sin ϴ The rectangular coordinates (2 √ 3,2) � If (r, ϴ ) in polar coordinates of the coordinates is given 2. Find the rectangular coordinates for the point with polar point are (x,y) coordinates (-4, 5 π /6) � 3. Sketch a graph of the polar curve r = sin ϴ - cos ϴ � ( x , y ) r 4. Sketch a graph of the polar curve r = cos(5 ϴ ) y θ x

Polar curves Match the curve with the A polar curve is a curve described by an a equation in equation � polar coordinates. � ❖ r = 3 cos(2 ϴ ) � Plot the following examples � ❖ r = (1-cos ϴ ) � ❖ r = 3 cos(2 ϴ ) � ❖ r= ϴ � ❖ r = (1-cos ϴ ) � ❖ r = sin ϴ ❖ r= ϴ � ❖ r = sin ϴ Area of a sector of a circle of radius r Consider the polar curve of equation � r = a (1-cos ϴ ) � (a is constant) ϴ in [0,2 π ] � 2 r A θ θ = A ⇒ = � 2 2 2 r π π (This curve is called cardiod . The animation is from Wolfram � http://mathworld.wolfram.com/ Area A of a region bounded by a polar curve Cardioid.html) of equation r=f( ϴ ), ϴ in [a,b] This curve has parametric equations � x=a cos(t)(1-cos(t)) � Example: Compute the area y=sin(t)(1-cos(t)) � bounded by the cardiod � ∆ θ r 2 / 2 = ∆ θ .f ( θ ) 2 / 2 � r = (1-cos ϴ ) � and its rectangular coordinates its points ϴ in [0,2 π ] R b a f ( θ ) 2 d θ A = 1 satisfy the equation � 2 (x 2 +y 2 +ax) 2 =a 2 (x 2 +y 2 )

❖ Find the area inside one petal of ❖ Find the area enclosed by the the curve r=3cos(2 ϴ ) curve r=sin( ϴ ) ❖ Find the area enclosed by the curve Arc length review r= ϴ , ϴ in [0,2 π ] and the positive x-axis (a,b) s s Δ k y Δ Δ y k (c,d) a x b Δ k Δ x

x r cos Length of a curve in polar coordinates " = θ & r f ( ) = θ Example: Express (but not # y r sin = θ $ evaluate) as an integral the arc length of the cardiod lying above Given the polar curve the x-axis. � r = 1-cos ϴ � ϴ in [0,2 π ] Differentiate with respect to ϴ Example: Express (but not Square and add evaluate) as an integral the length of a petal of 3 cos(2 ϴ ) � Using the formula for parametric curves Example: Compute the arc length of Example: Compute the arc length the the circle r = sin ϴ � graph of the curve y=x 3/2 , � ϴ in [0,2 π ] x in [0,5] Example: Compute the arc length of the circle r = sin ϴ � ϴ in [0,2 π ] POLAR FUNCTION Example: Compute the length arc of Example: Compute the length two Example: Compute the length arc arcs of the cycloid x= ϴ - sin ϴ , y=1- the spiral r = e ϴ � of the spiral r = e ϴ � cos ϴ . (Hint (1-cos(t))=2 sin 2 (t/2)) in [0,2 π ] in [0,2 π ] PARAMETRIC POLAR