Lecture Note Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 10.2 Plane Curves and Parametric Equations 10.3 Parametric Equations and Calculus 10.4 Polar Coordinates and Polar Graphs 10.5 Area and Arc Length in Polar Coordinates 10.6 Polar Equations of Conics and Kepler’s Laws Lecture Note 10.4 Polar Coordinates and Polar Graphs Lecture Note 10.4 Polar Coordinates and Polar Graphs

Polar Coordinates A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. Usually we use Cartesian coordinates, which are directed distances from two perpendicular axes. Here we describe a coordinate system introduced by Newton, called the polar coordinate system. 푃(푟, 휃) We choose a point in the plane that is called the pole (or origin) and is labeled 푂. Directed distance 푟 Then we draw a ray (half-line) starting at 푂 called the polar 휃 (directed angle) axis. This axis is usually drawn horizontally to the right and 푂 polar axis corresponds to the positive 푥-axis in Cartesian coordinates. pole If 푃 is any other point in the plane, let 풓 be the directed distance from 푶 to 푷 and let 휽 be the directed angle (usually measured in radians: positive if measured in the counterclockwise from the polar axis and negative in the clockwise direction) between the polar axis and the line 푶푷. Then the point 푃 is represented by the ordered pair (푟, 휃) and 푟, 휃 are called polar coordinates of 푃. For convenience, we allow 푟 to be negative, with the understanding that in this case, the point is in the opposite direction from that indicated by the angle 휃. Lecture Note 10.4 Polar Coordinates and Polar Graphs

The following figure shows three points on the polar coordinate system, it is convenient to locate points with respect to a grid of concentric circles intersected by radial lines through the pole. Lecture Note 10.4 Polar Coordinates and Polar Graphs

Uniqueness Issue on Polar Coordinates Representation With rectangular coordinates, each point (푥, 푦) has a unique representation. This is not true with polar coordinates. The polar coordinates 푟, 휃 , 푟, 휃 + 2푛휋 , −푟, 휃 + 2푛 + 1 휋 , 푛 is any integer represent the same point. Moreover, the pole is represented by (0, 휃), where 휃 is any angle. Lecture Note 10.4 Polar Coordinates and Polar Graphs Coordinate Conversion between Cartesian coordinates and Polar Coordinates The connection between polar and Cartesian coordinates can be shown from the figure below, in which the pole corresponds to the origin and the polar axis coincides with the positive 푥-axis. If the point 푃 has Cartesian coordinates (푥, 푦) and polar coordinates (푟, 휃), then from the figure, we have 푦 푥 푦 cos 휃 = sin 휃 = 푃 푟, 휃 = 푃(푥, 푦) 푟 푟 and so 푥 = 푟 cos 휃 푦 = 푟 sin 휃 푟 푦 휃 polar axis 푦 푟2 = 푥2 + 푦2 tan 휃 = 푂 푥 푥 푥 pole Lecture Note 10.4 Polar Coordinates and Polar Graphs Example 1 Plot each point given by polar coordinates and find its rectangular coordinates. 휋 a 푟, 휃 = 2, 휋 b −2, 휋 c 3, 6

Example 2 Plot each point given by rectangular coordinates and find its polar coordinates. a −1, 1 b 0, 2 Lecture Note 10.4 Polar Coordinates and Polar Graphs

Polar Graph The graph of a polar equation 푟 = 푓(휃), called a polar curve, is the set of all points (푥, 푦) for which 푥 = 푟 cos 휃 , 푦 = 푟 sin 휃 and 푟 = 푓 휃 . In other words, the graph of a polar equation is a graph in the 푥푦-plane of all those points whose polar coordinates satisfy the given equation.

Two general methods of sketching the graph of a polar equation by hand: 1) By plotting sample points and connecting them with smooth curves 2) By converting a polar equation to a rectangular equation. 3) By sketching the graph of a polar equation on 휃푟-plane, then apply on 푥푦-plane.

Note when a polar graph is sketched by a graphing calculator which does not have a polar mode, we can graph 푟 = 푓(휃) by writing the equation as parametric equation with 휃 as a parameter 푥 = 푟 cos 휃 = 푓 휃 cos 휃 and 푦 = 푟 sin 휃 = 푓 휃 sin 휃 Lecture Note 10.4 Polar Coordinates and Polar Graphs

Example 3 Describe the graph of each polar equation. Confirm each description by converting to a rectangular equation. 휋 a 푟 = 2 b 휃 = c 푟 = sec 휃 3 Lecture Note 10.4 Polar Coordinates and Polar Graphs

Example 4 Sketch the graph of 푟 = 2 cos 3휃 . 10.4 Polar Coordinates and Polar Graphs Lecture Note

Review of Polar Equations Circles in Polar Coordinates The graph of 푟 = 푎 cos 휃 and 푟 = 푎 sin 휃, 푎 > 0 are circle.

푎 • 푟 = 푎 cos 휃: circle with radius 푎/2 centered at , 0 . 2 푎 • 푟 = 푎 sin 휃: circle with radius 푎/2 centered at 0, . 2 10.4 Polar Coordinates and Polar Graphs Lecture Note Review of Polar Equations

Limicons in Polar Coordinates The graph of 푟 = 푎 + 푏 sin 휃 , 푟 = 푎 − 푏 sin 휃 푟 = 푎 + 푏 cos 휃 , 푟 = 푎 − 푏 cos 휃, 푎 > 0, 푏 > 0 are called limacons, The ratio 푎/푏 determines a limacon’s shape. 10.4 Polar Coordinates and Polar Graphs Lecture Note

Review of Polar Equations

Rose Curves in Polar Coordinates The graphs of 푟 = 푎 sin 푛휃 and 푟 = 푎 cos 푛휃 , 푎 ≠ 0, are called rose curves. If 푛 is even, the rose has 2푛 petals. If 푛 is odd, the rose has 푛 petals. 10.4 Polar Coordinates and Polar Graphs Lecture Note Review of Polar Equations

Lemniscates in Polar Coordinates The graphs of 푟2 = 푎2 sin 푛휃 and 푟2 = 푎2 cos 푛휃 , 푎 ≠ 0, are called lemniscates. 10.4 Polar Coordinates and Polar Graphs Lecture Note Lecture Note 10.4 Polar Coordinates and Polar Graphs Theorem 10.11 Tangents to Polar Curves To find a tangent line to a polar curve 푟 = 푓(휃), we regard 휃 as a parameter and write its parametric equations as 푥 = 푟 cos 휃 = 푓 휃 cos 휃 푦 = 푟 sin 휃 = 푓 휃 sin 휃 Then, using the method for finding slopes of parametric curves and the Product Rule, we have 푑푦 푑 푑푟 푑 푑푦 푟 sin 휃 sin 휃 + 푟 sin 휃 푓′ 휃 sin 휃 + 푓 휃 cos 휃 = 푑휃 = 푑휃 = 푑휃 푑휃 = 푑푥 푑 푑푟 푑 ′ 푑푥 (푟 cos 휃) cos 휃 + 푟 cos 휃 푓 휃 cos 휃 − 푓 휃 sin 휃 푑휃 푑휃 푑휃 푑휃 Lecture Note 10.4 Polar Coordinates and Polar Graphs Theorem 10.11 Tangents to Polar Curves To find a tangent line to a polar curve 푟 = 푓(휃), we regard 휃 as a parameter and write its parametric equations as 푥 = 푟 cos 휃 = 푓 휃 cos 휃 푦 = 푟 sin 휃 = 푓 휃 sin 휃 Then, using the method for finding slopes of parametric curves and the Product Rule, we have 푑푦 푑 푑푟 푑 푑푦 푟 sin 휃 sin 휃 + 푟 sin 휃 푓′ 휃 sin 휃 + 푓 휃 cos 휃 = 푑휃 = 푑휃 = 푑휃 푑휃 = 푑푥 푑 푑푟 푑 ′ 푑푥 (푟 cos 휃) cos 휃 + 푟 cos 휃 푓 휃 cos 휃 − 푓 휃 sin 휃 푑휃 푑휃 푑휃 푑휃 We locate 푑푦 푑푥 . horizontal tangents by finding the points where = 0 (provided that ≠ 0). 푑휃 푑휃 Likewise, we locate 푑푥 푑푦 • vertical tangents at the points where = 0 (provided that ≠ 0). 푑휃 푑휃 If 푑푦/푑휃 and 푑푥/푑휃 are simultaneously 0, then no conclusion can be drawn about tangent lines. Lecture Note 10.4 Polar Coordinates and Polar Graphs Theorem 10.11 Tangents to Polar Curves To find a tangent line to a polar curve 푟 = 푓(휃), we regard 휃 as a parameter and write its parametric equations as 푥 = 푟 cos 휃 = 푓 휃 cos 휃 푦 = 푟 sin 휃 = 푓 휃 sin 휃 Then, using the method for finding slopes of parametric curves and the Product Rule, we have 푑푦 푑 푑푟 푑 푑푦 푟 sin 휃 sin 휃 + 푟 sin 휃 푓′ 휃 sin 휃 + 푓 휃 cos 휃 = 푑휃 = 푑휃 = 푑휃 푑휃 = 푑푥 푑 푑푟 푑 ′ 푑푥 (푟 cos 휃) cos 휃 + 푟 cos 휃 푓 휃 cos 휃 − 푓 휃 sin 휃 푑휃 푑휃 푑휃 푑휃

Theorem 10.12 Tangents to Polar Curves 푑푦 If 푓 훼 = 0 and 푓′ 훼 ≠ 0, then = tan 훼 so that the line 휃 = 훼 is tangent at the pole 푑푥 to the graph of 푟 = 푓(휃). Lecture Note 10.4 Polar Coordinates and Polar Graphs

Example 5 Find the horizontal and vertical tangent lines of 푟 = sin 휃 , 0 ≤ 휃 ≤ 휋. Lecture Note 10.4 Polar Coordinates and Polar Graphs

Example 6 Find the horizontal and vertical tangent lines of 푟 = 2(1 − cos 휃).

End of Chapter 10.4 Lecture Note 10.5 Area and Arc Length in Polar Coordinates Theorem 10.13 Area in Polar Coordinates If 푓 is continuous and nonnegative on the interval [훼, 훽], 0 < 훽 − 훼 ≤ 2휋, then the area of the region bounded by the graph of 푟 = 푓(휃) between two radian lines 휃 = 훼 and 휃 = 훽 is 훽 1 훽 1 퐴 = 푓 휃 2 푑휃 = 푟2 푑휃 훼 2 훼 2

푟 = 푓(휃) 푟 = 푓(휃)

훽 훽 훼 훼 Lecture Note 10.5 Area and Arc Length in Polar Coordinates

Example 1 Find the area of one petal of the rose curve 푟 = 3 cos(3휃) . Lecture Note 10.5 Area and Arc Length in Polar Coordinates Example 2 Find the area of the region lying between the inner and outer lops of the limacon 푟 = 1 − 2 sin 휃 . Lecture Note 10.5 Area and Arc Length in Polar Coordinates Example 3 Find the area of the region common to the two regions bounded by the curves 푟 = −6 cos 휃 and 푟 = 2 − 2 cos 휃 . Lecture Note 10.5 Area and Arc Length in Polar Coordinates

Points of Intersection of Polar Graphs Because a point may be represented in different ways in polar coordinates, care must be taken in determining the points of intersection of two polar graphs.

For example, consider the points of intersection of the graphs of 푟 = 1 − 2 cos 휃 푎푛푑 푟 = 1. As with rectangular equations, you can attempt to find the points of intersection by solving the two equations simultaneously, as shown 휋 3휋 푟 = 1 − 2 cos 휃 = 1 ⇒ 휃 = , . 2 2 However, if you can sketch the graphs of two polar curves, you will see there is a third point of intersection that did not show up when the two polar equations were solved simultaneously.

The reason the third point was not found is that it does not occur with the same coordinates in the two graphs. On the graph of 푟 = 1, the point occurs with coordinates (1, 휋), but on the graph of 푟 = 1 − 2 cos 휃 the point occurs with coordinates (−1,0). So in finding intersection of polar curves solve equations simultaneously and also sketch their graphs. In addition to that, note that because the pole can be represented by (0, 휃), where 휃 is any angle, you should check separately for the pole when finding points of intersection. Lecture Note 10.5 Area and Arc Length in Polar Coordinates Example 3 Find the area of the region common to the two regions bounded by the curves 푟 = −6 cos 휃 and 푟 = 2 − 2 cos 휃 . Lecture Note 10.5 Area and Arc Length in Polar Coordinates

Theorem 10.14 The length of a polar curve Let 푓 be a function whose derivative is continuous on an interval 훼 ≤ 휃 ≤ 훽. The length of the graph of 푟 = 푓(휃) from 휃 = 훼 to 휃 = 훽 is 훽 푟 = 푓(휃) 푠 = 푥′ 휃 2 + 푦′ 휃 2푑휃 훼 훽 2 ′ 2 = [푓 휃 + 푓 휃 푑휃 훽 훼 2 훼 훽 푑푟 = 푟2 + 푑휃 훼 푑휃

When applying the arc length formula to a polar curve, be sure that the curve is traced out only once on the interval of integration. For instance, the rose curve 푟 = cos 3휃 is traced out once on the interval 0 ≤ 휃 ≤ 휋, but is traced out twice on the interval 0 ≤ 휃 ≤ 2휋. Lecture Note 10.5 Area and Arc Length in Polar Coordinates Example 4 Find the length of the arc from 휃 = 0 to 휃 = 2휋 for the cardioid 푟 = 푓 휃 = 2 − 2 cos 휃 . Lecture Note 10.5 Area and Arc Length in Polar Coordinates

Theorem 10.15 Area of a Surface of Revolution 푟 = 푓 휃 from 휃 = 훼 to 휃 = 훽 is Let 푓 be a function whose derivative is continuous on an interval 훼 ≤ 휃 ≤ 훽. The area of the surface formed by revolving the graph of 푟 = 푓(휃) from 휃 = 훼 to 휃 = 훽 about the indicated line is as follows 훽 푆 = 2휋 푓 휃 sin 휃 푓 휃 2 + 푓′ 휃 2푑휃 About the polar axis 훼 훽 휋 푆 = 2휋 푓 휃 cos 휃 푓 휃 2 + 푓′ 휃 2푑휃 About the line 휃 = 훼 2 Lecture Note 10.5 Area and Arc Length in Polar Coordinates Example 5 Find the area of the surface formed by revolving the circle 푟 = 푓 휃 = cos 휃 about the line 휃 = 휋/2.

End of Chapter 10.5