Chapter 16

August 17 2017

Chapter 16 Polar Coordinates 16.1 The Polar Coordinate System - a point is located by means of its distance and direction from the origin - polar coordinates (r, angle) - direction angle = 0, the positive x-axis, is called the polar axis - it is important to know the connection between rectangular and polar coordinates - x = r cos angle, y = r sin angle - just as in the case of rectangular coordinates - the graph of a polar equation - F (r, angle) = 0 - is the set of all points P = (r, angle) whose polar coordinates satisfies the equation - in most situations we will encounter - the equation can be solved for r, and takes the form - r = f(angle) - we merely choose a convenient sequence of values for angle - each determining its own direction from the origin - and compute the corresponding values of r that tell us how far to go in each of these directions 16.2 More Graphs of Polar Equations - the curve - r = a (1 + cos angle) with a > 0 - is a cardioid - the curve r = a (1 + 2 cos angle) with a > 0 - is a limacon (snail) - r2 = 2a2 cos 2 angle is a lemniscate (figure eight) 16.3 Polar Equations of Circles, Conics, and Spirals - we had translated polar coordinates directly to rectangular coordinates - another way is to derive the polar coordinates from some characteristic geometric property of the curve - polar coordinates are particularly well suited for working with conic sections 16.4 Arc Length and Tangent Lines - we can use integration to find the length of arc lengths in polar coordiantes - the formula for ds can also be used to find areas of the surfaces of revolution 16.5 Areas in Polar Coordinates - we think of the total area A as a result of adding up elements of area dA
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}