MATH115. Polar Coordinate System and Polar Graphs. Paolo Lorenzo Bautista. June 14, De La Salle University

Transcription

1 MATH115 Polar Coordinate System and Paolo Lorenzo Bautista De La Salle University June 14, 2014 PLBautista (DLSU) MATH115 June 14, / 30

2 Polar Coordinates and PLBautista (DLSU) MATH115 June 14, / 30

3 Polar Coordinates and Example Locate the following points given in their polar coordinates: 1. (2, 1 4 π) 2. (5, 1 2 π) PLBautista (DLSU) MATH115 June 14, / 30

4 Polar Coordinates and Example Locate the following points given in their polar coordinates: 1. (2, 1 4 π) 2. (5, 1 2 π) Remark 1. A point can have infinitely many polar coordinates. 2. r can be negative. 3. The coordinates of a point P = (r, θ) is unique if r > 0 and 0 θ < 2π. PLBautista (DLSU) MATH115 June 14, / 30

5 Cartesian to Polar, and vice versa Polar Coordinates and 1. To convert a point P = (r, θ) to Cartesian form, we use the following equations: x = r cos θ y = r sin θ PLBautista (DLSU) MATH115 June 14, / 30

6 Cartesian to Polar, and vice versa Polar Coordinates and 1. To convert a point P = (r, θ) to Cartesian form, we use the following equations: x = r cos θ y = r sin θ 2. To convert a point P = (x, y) to polar form, we use the following equations: tan θ = y x r = x 2 + y 2 PLBautista (DLSU) MATH115 June 14, / 30

7 Exercise Polar Coordinates and Example 1. Convert the following points to polar form. (3, 4) ( 5, 12) ( 12, 5) (1, 1) ( 3, 1) 2. Convert the following points to Cartesian form. (4, π 3 ) (3, 5π 4 ) ( 2, 3π 4 ) PLBautista (DLSU) MATH115 June 14, / 30

8 Polar Coordinates and PLBautista (DLSU) MATH115 June 14, / 30

9 Polar Coordinates and Example For the following polar equations, find their corresponding Cartesian equation. 1. r 2 = 4 sin 2θ 2. r 2 cos 2θ = r 2 = 4 cos 2θ 4. r = 2 sin 3θ PLBautista (DLSU) MATH115 June 14, / 30

10 Lines Definition A line is the graph of any of the following equations: 1. θ = C 2. θ = C + kπ 3. r sin θ = b 4. r sin θ = a PLBautista (DLSU) MATH115 June 14, / 30

11 Circles Definition A circle is the graph of any of the following equations: 1. r = C 2. r = C 3. r = 2a cos θ 4. r = 2b sin θ PLBautista (DLSU) MATH115 June 14, / 30

12 Tests for Symmetry If a point P = (r, θ) is on a polar graph, then the graph is 1. symmetric with respect to the polar axis if (r, θ) or ( r, π θ) are also on the graph. 2. symmetric with respect to the π 2 -axis if (r, π θ) or ( r, θ) are also on the graph. 3. symmetric with respect to the pole if ( r, θ) or (r, π + θ) are also on the graph. PLBautista (DLSU) MATH115 June 14, / 30

13 Limacons Definition A limacon is the graph of an equation of the form r = a ± b cos θ or r = a ± b sin θ, where a > 0 and b > 0. PLBautista (DLSU) MATH115 June 14, / 30

14 Limacons Definition A limacon is the graph of an equation of the form r = a ± b cos θ or r = a ± b sin θ, where a > 0 and b > 0. Four types: 1. 0 < a b < 1 Limacon with a loop a 2. b = 1 Cardioid 3. 1 < a b < 2 Limacon with a dent a 4. b 2 Convex limacon PLBautista (DLSU) MATH115 June 14, / 30

15 Example Sketch the following graphs: 1. r = 1 2 cos θ 2. r = sin θ 3. r = cos θ 4. r = 2 sin θ PLBautista (DLSU) MATH115 June 14, / 30

16 Limacon with a loop r = 1 2 cos θ PLBautista (DLSU) MATH115 June 14, / 30

17 Limacon with a dent r = sin θ PLBautista (DLSU) MATH115 June 14, / 30

18 Cardioid r = cos θ PLBautista (DLSU) MATH115 June 14, / 30

19 Convex limacon r = 2 sin θ PLBautista (DLSU) MATH115 June 14, / 30

20 Rose Definition A rose is the graph of an equation of the form r = a cos nθ or r = a sin nθ, where n is a positive integer. If n is even, the rose has 2n leaves; if n is odd, the rose has n leaves. PLBautista (DLSU) MATH115 June 14, / 30

21 Rose Definition A rose is the graph of an equation of the form r = a cos nθ or r = a sin nθ, where n is a positive integer. If n is even, the rose has 2n leaves; if n is odd, the rose has n leaves. Sketch the following graphs: 1. r = 4 cos 2θ 2. r = 4 cos 3θ 3. r = 2 sin 2θ PLBautista (DLSU) MATH115 June 14, / 30

22 r = 4 cos 2θ PLBautista (DLSU) MATH115 June 14, / 30

23 r = 4 cos 3θ PLBautista (DLSU) MATH115 June 14, / 30

24 r = 2 sin 2θ PLBautista (DLSU) MATH115 June 14, / 30

25 Length of Arc and Areas of a Region Length of Arc Theorem Let C be a curve with polar equation r = F(θ) where F (θ) is continuous on the closed interval [α, β]. Then the length of the arc of the curve C is given by L = β α ( ) dr 2 + r dθ 2 dθ. PLBautista (DLSU) MATH115 June 14, / 30

26 Length of Arc and Areas of a Region Example Find the length of the arcs of the following polar graphs. 1. r = 2(1 + cos θ) 2. r = 4 sin θ 3. r = sin θ from θ = 0 to θ = 1 2 π PLBautista (DLSU) MATH115 June 14, / 30

27 Length of Arc and Areas of a Region Exercise Find the length of the arcs of the following polar graphs. 1. r = 1 sin θ 2. r = 3θ from θ = 0 to θ = 2π 3. r = 3 cos θ PLBautista (DLSU) MATH115 June 14, / 30

28 Finding the Points of Intersection Length of Arc and Areas of a Region Example Determine the number of points of intersection of the following graphs by sketching. { r = 2 cos θ 1. r = 2 sin θ { r = 2 sin 2θ 2. r = 1 PLBautista (DLSU) MATH115 June 14, / 30

29 Length of Arc and Areas of a Region Finding the Points of Intersection Remark If a polar graph has an equation r = f (θ), then the same curve is given by ( 1) n r = f (θ + nπ) where n is an integer. PLBautista (DLSU) MATH115 June 14, / 30

30 Length of Arc and Areas of a Region Finding the Points of Intersection Remark If a polar graph has an equation r = f (θ), then the same curve is given by ( 1) n r = f (θ + nπ) where n is an integer. Steps in finding the points of intersection of r = f (θ) and r = g(θ): 1. Find all distinct equations of r = f (θ) and r = g(θ). 2. Solve all each of the equations of r = f (θ) simultaneously with the equations of r = g(θ). 3. Check to see if the pole is a point of intersection. PLBautista (DLSU) MATH115 June 14, / 30

31 Length of Arc and Areas of a Region Example Find the points of intersection for the following graphs { r = 2 cos θ r = 2 sin θ { r = 2 sin 2θ r = 1 { r = 2 cos 2θ 3. r = 2 sin θ { r = 2 cos 2θ 4. r = 2 sin θ PLBautista (DLSU) MATH115 June 14, / 30

32 Areas of Regions Length of Arc and Areas of a Region Theorem Let R be the region bounded by the lines θ = α and θ = β and the curve whose equation is r = f (θ), where f is continuous and nonnegative on the closed interval [α, β]. If A square units is the area of region R, then A can be computed as A = 1 2 β α [f (θ)] 2 dθ. PLBautista (DLSU) MATH115 June 14, / 30

33 Length of Arc and Areas of a Region Example Find the area of the region covered by the following graphs. 1. r = cos θ 2. r = 2 sin θ 3. r = 4 sin θ 4. One loop of r = 4 sin 3θ PLBautista (DLSU) MATH115 June 14, / 30

34 Areas of Regions Length of Arc and Areas of a Region Theorem Let R be the region bounded by the lines θ = α and θ = β and two curves whose equation are given by r = f (θ) and r = g(θ), where f and g are continuous on the closed interval [α, β] and f (θ) g(θ) on [α, β]. If A square units is the area of region R, then A can be computed as A = 1 2 β α ([f (θ)] 2 [g(θ)] 2 )dθ. PLBautista (DLSU) MATH115 June 14, / 30

35 Length of Arc and Areas of a Region Example Find the area of the specified regions. 1. The intersection of the regions covered by r = 2 and r = 3 2 cos θ 2. The intersection of the regions covered by r = 4 sin θ and r = 4 cos θ 3. The intersection of the regions covered by r = 3 sin 2θ and r = 3 cos 2θ 4. The region inside r = 3 and outside r = 3(1 cos θ) 5. The region inside r 2 = 4 sin 2θ and outside r = 2 PLBautista (DLSU) MATH115 June 14, / 30