10.1 Review of Parametric Equations

Transcription

1 10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations using a parameter such as t. This means that x and y are both separately defined as functions of this parameter, x = f(t) and y = g(t). If you are given parametric equations, you can sometimes create a Cartesian equation by eliminating the parameter. How? If possible, solve one of the parametric equations for t and use substitution. If the parametric equations involve trig functions, use a trig identity. Sometimes, there may be a restriction on the values of t or the values of x and y may have bounds you need to watch out for. Example: Eliminate the parameter to find a Cartesian equation for the following curves, sketch a graph, and denote the direction that the curve is traced out as t increases. x = t + 1, y = t 3 x = t 2, y = t 2, 2 t 4 x = t, y = 2 t 1

2 x = 2 cos θ, y = 2 sin θ x = 3 + sin t, y = 1 + cos t x = 4 cos θ, y = 5 sin θ, 0 θ π x = cos t, y = sec t, 0 t π 3 2

3 10.2 Arc Length and Surface Area with Parametric Curves Given a parametric curve defined by x = f(t), y = g(t), the length of the curve from t = a to t = b is: L = b a [f (t)] 2 + [g (t)] 2 dt or b a (dx ) 2 + dt ( ) dy 2 dt dt Examples: Find the length of the curve x = t 2 + 4, y = t 3 + 1, from the point (4, 1) to the point (8, 9). 3

4 Find the length of the curve x = 4e t 4t, y = 16e t/2, 0 t 1. Surface Area: The surface area of a solid of revolution where a curve x = x(t), y = y(t) with arc length ds is rotated about an axis is (dx ) 2 ( ) dy 2 S = 2πr ds = 2πr + dt dt dt If revolving about the x-axis, r = y and if revolving about the y-axis, r = x. 4

5 Examples: Find the surface area of the solid obtained by rotating the curve x = 3t 3 t, y = 3t 2, 0 t 1, about the x-axis. Find the surface area of the solid formed by rotating the curve x = e 3t + e 3t, y = 10 6t, 0 t 2, about the y-axis. 5

6 Section 10.3 Polar Coordinates The polar coordinate system is a new way of thinking about graphing curves. Instead of the rectangular coordinate system (x, y), polar coordinates are of the form (r, θ) where θ is the angle made with the positive x-axis and r is the distance from the origin. Examples: Plot the following points given in polar coordinates (r, θ). A. (2, π/3) B. (3, 5π/6) C. ( 4, 3π/4) To convert from polar to rectangular coordinates, we use the following: x = r cos θ y = r sin θ Find the Cartesian coordinates of the polar points plotted above. (a) (2, π/3) (b) (3, 5π/6) (c) ( 4, 3π/4) 6

7 Going from Cartesian to polar coordinates, we use the following facts: r 2 = x 2 + y 2 tan θ = y x However, the way we write a point in polar coordinates is NOT unique. Convert the following points with Cartesian coordinates into polar coordinates in three different ways. ( 1, 3) (i) r > 0, 0 θ 2π (ii) r < 0, 0 θ 2π (iii) r > 0, 2π θ 0 ( 3, 3) Find a polar equation for the curve represented by the following Cartesian equations. x 2 + y 2 = 25 x 2 + y 2 = 6y y = x 2 7

8 Find a Cartesian equation for the following polar curves. r = 2 cos θ r = 4 csc θ Polar equations are usually written where r is a function of θ, i.e. as r = f(θ). Sketch graphs of the following polar equations. r = 3 θ = π/6 Sketch the region 2 r < 5, 2π/3 θ < 7π/6 8

9 r = 3 cos θ r = 4 sin θ Summary of Circles: r = a r = a cos θ, a > 0 r = a cos θ, a < 0 r = a sin θ, a > 0 r = a sin θ, a < 0 9

10 Cardioids are polar graphs of the form r = a ± a cos θ or r = a ± a sin θ. r = sin θ r = 2 2 cos θ Summary of Cardioids r = a + a cos θ, a > 0 r = a a cos θ, a > 0 r = a + a sin θ, a > 0 r = a a sin θ, a > 0 10

11 Roses are polar graphs of the form r = a sin(kθ) or r = a cos(kθ). r = sin(2θ) r = cos(3θ) Summary of Roses: r = cos(kθ), k odd, k petals r = cos(kθ), k even, 2k petals r = sin(kθ), k odd, k petals r = sin(kθ), k even, 2k petals 11

12 10.4 Areas and Lengths in Polar Coordinates Given a polar equation r = f(θ), the area bounded by the curve between θ = a and θ = b is given by A = b a 1 b 1 2 r2 dθ = a 2 [f(θ)]2 dθ Find the area of the region bounded by r = e 2θ within the sector 0 θ π/2. 12

13 Find the area of the region bounded by the circle r = 3 cos θ within the sector π/6 θ π/3. Find the area inside the cardioid r = sin θ. Find the area of one loop (petal) of the rose r = 3 sin(5θ). 13

14 Set up an integral to find the area of one loop (petal) of the rose r = 2 cos(4θ). The area of the region bounded by two polar curves r 1 and r 2 where r 1 is the inner polar curve and r 2 is the outer polar curve is given by A = b a 1 2 [r2 2 r 2 1] dθ Find the area inside the circle r = 6 sin θ and oustide the circle r = 3. 14

15 Find the area inside the circle r = 6 cos θ and outside the cardioid r = cos θ. The arc length of a polar curve r = f(θ), a θ b is given by L = b a r 2 + ( ) dr 2 dθ dθ Find the length of the polar curve r = θ 2, 0 θ π/2. Find the length of the polar curve r = e 3θ, 0 θ π. 15

16 Conic Sections in Cartesian and Polar Coordinates There are three types of conic sections: parabolas, ellipses, and hyperbolas. (Keep in mind a circle is a special case of an ellipse.) The general equation of a parabola with vertex (h, k) that opens up/down is given below. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward. y = a(x h) 2 + k The general equation of a parabola with vertex (h, k) that opens right/left is given below. If a is positive, the parabola opens to the right. If a is negative, the parabola opens to the left. x = a(y k) 2 + h Find the vertex of the parabola and sketch a graph. x 3 = 2(y + 3) 2 The general form of an ellipse centered at (h, k) is given by: (x h) 2 (y k)2 a 2 + b 2 = 1 Sketch a graph of the ellipse 16x 2 + 4y 2 = 64 by first finding x and y-intercepts. Find the center of the ellipse 3x 2 + y x 10y =

17 The general equation of a hyperbola centered at the origin is x 2 a 2 y2 b 2 = 1 or y 2 b 2 x2 a 2 = 1 Sketch a graph of the hyperbola x2 36 y2 = 1 by finding intercepts. 25 Sketch a graph of the hyperbola 9y 2 x 2 = 81 by finding intercepts. Identify the type of conic given by the following equations. x 2 3x = 6y y 2 x + 3 = y 2 + y 5x 2 + 6y = 3x + 4y 2 17

18 Conic sections can also be represented in polar form. The general form of a conic in polar form is one of the following: a r = 1 ± e cos(θ) or r = a 1 ± e sin(θ) The number e is called the eccentricity and determines which type of conic the polar equation represents. If e < 1, the conic is an ellipse. If e = 1, the conic is a parabola. If e > 1, the conic is a hyperbola. Notes: The number a is a constant that has geometric meaning (which we don t discuss). The ± and the cos(θ) vs. sin(θ) determine the general shape and properties of the conic. We won t get into that in this class, but you can see how these affect the shape by converting from a polar equation to a Cartesian equation and completing the square. Identify the conic represented by the polar equations below. r = cos(θ) r = sin(θ) r = cos(θ) 18