Area of an Oﬀ Center Circle Let’s ﬁnd the area in polar coordinates of the region enclosed by the curve r = 2a cos θ. We’ve previously shown that this curve describes a circle with radius a centered at (a, 0). In rectangular coordinates its equation is (x − a)2 + y 2 = a2 . y r θ x (a,0) Figure 1: Oﬀ center circle r = 2a cos θ. We’re going to integrate an inﬁntessimal amount of area dA. The integral will go from θ1 = − π2 to θ2 = π2 . We could ﬁnd these limits by looking at Figure 1; to draw the circle we might start by moving “down” at angle − π2 . As we move along the bottom of the circle toward (2a, 0) the angle increases to 0, and as we trace out the top of the circle we’re moving from angle 0 to angle π2 (“up”). We might also ﬁnd the limits of integration by looking at the formula and realizing that the cosine function is positive for − π2 < θ < π2 . When θ = ± π2 , r = 2a cos θ is 0, so the two ends of the curve meet at the origin. Our inﬁnitessimal unit of area is dA = 12 r2 dθ, so: π 2 � A = −π 2 π 2 � = −π 2 π 2 � 1 2 r dθ 2 1 (2a cos θ)2 dθ 2 2a2 cos2 θ dθ = −π 2 = 2 � π 2 2a cos2 θ dθ (half angle formula) −π 2 = � π 2 1 + cos 2θ dθ 2 � � π2 1 2 a θ + sin 2θ 2 −π 2 2a −π 2 = 2 �π = a2 = πa2 . � π �� − − 2 2 We know that the area of a circle of radius a is πa2 ; our answer is correct.