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# polar - POLAR INTEGRATION HOMEWORK 14.5 8,10 30,54,60...

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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 , θ = arctg( y/x ). We have x = r cos( θ ), y = r sin( θ ). 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 ) , θ ( t )). It can be translated into x, y coordinates: x ( t ) = r ( t ) cos( θ ( t ) , y ( t ) = r ( t ) sin( θ ( t )). POLAR GRAPHS. Curves which are graphs when written in polar coordinates are called polar graphs . EXAMPLE. r ( θ ) = cos(3 θ ) 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( θ ) = 2 r cos( θ ) + 3. Solving for r ( t ) gives r ( θ ) = 3 / (sin( θ 2 cos( θ )). The line is also a polar graph. EXAMPLE. The polar form r ( θ ) = a (1 - ǫ 2 ) 1+ ǫ cos( θ ) of the ellipse (see Kepler). The ellipse is a polar graph. INTEGRATION IN POLAR COORDINATES. For many regions, it is better to use polar coordinates for integration: integraltext integraltext f ( x, y ) dxdy = integraltext integraltext g ( r, θ ) r drdθ For example if f ( x, y ) = x
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