POLAR INTEGRATIONMath21a, O. KnillHOMEWORK: 14.5: 8,10, 30,54,60REMINDER:POLARCOORDINATES.Apoint(x, y)intheplanehasthepolar coor-dinatesr=radicalbigx2+y2, θ= arctg(y/x).We havex=rcos(θ),y=rsin(θ).xyP=(x,y)=(r cos(t),r sin(t))O=(0,0)r=d(P,O)tPOLAR CURVES. A general polar curve is written as (r(t), θ(t)). It can be translated intox, ycoordinates: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 calledpolar graphs.EXAMPLE.r(θ) = cos(3θ) is thewhich belongs to the class ofrosesr(t) = cos(nt).EXAMPLE. Ify= 2x+ 3 is a line, then the equation givesrsin(θ) = 2rcos(θ) + 3.Solving forr(t) givesr(θ) = 3/(sin(θ−2 cos(θ)). The line is also a polar graph.EXAMPLE. The polar formr(θ) =a(1-ǫ2)1+ǫcos(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 forintegration:integraltext integraltextf(x, y)dxdy=integraltext integraltextg(r, θ)r drdθFor example iff(x, y) =x
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