polar - POLAR INTEGRATION Math21a, O. Knill HOMEWORK: 14.5:...

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Unformatted text preview: POLAR INTEGRATION Math21a, O. Knill HOMEWORK: 14.5: 8,10, 30,54,60 REMINDER: POLAR COORDINATES. A point ( x, y ) in the plane has the polar coor- dinates r = radicalbig x 2 + y 2 , = arctg( y/x ). We have x = r cos( ), y = r sin( ). x y P=(x,y)=(r cos(t),r sin(t)) O=(0,0) r=d(P,O) t POLAR CURVES. A general polar curve is written as ( r ( t ) , ( t )). It can be translated into x, y coordinates: x ( t ) = r ( t )cos( ( t ) , y ( t ) = r ( t )sin( ( t )). POLAR GRAPHS. Curves which are graphs when written in polar coordinates are called polar graphs . EXAMPLE. r ( ) = cos(3 ) is the which belongs to the class of roses r ( t ) = cos( nt ). EXAMPLE. If y = 2 x + 3 is a line, then the equation gives r sin( ) = 2 r cos( ) + 3. Solving for r ( t ) gives r ( ) = 3 / (sin( 2 cos( )). The line is also a polar graph. EXAMPLE. The polar form r ( ) = a (1- 2 ) 1+ cos( ) of the ellipse (see Kepler). The ellipse is a polar graph....
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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