Graph of r = 2a cos θ Let’s get some more practice in graphing and polar coordinates. We just found the area enclosed by the curve r = 2a cos θ for − π2 ≤ θ ≤ π2 . What happens when θ doesn’t lie in this range? y r θ (a,0) x Figure 1: Oﬀ center circle r = 2a cos θ. √ When π2 < θ < π, r is negative. For example, when θ = 34π , cos θ = − 22 and √ √ r = −a 2. If we move a distance of negative a 2 in the direction of angle 34π √ 3π we arrive at the point (−a 2, 4 ), which is (a, −a) in rectangular coordinates. In fact, because we know that the points on the curve must have the property: (x − a)2 + y 2 = a2 in rectangular coordinates, we know that as θ increases, the point (2a cos θ, θ) must remain on that same curve. As θ ranges from 0 to 2π (or from − π2 to 32π ), the point (2a cos θ, θ) travels around the circle twice. A common mistake is to choose the wrong limits of integration and count the same area twice, or cancel a positive area with an overlapping negative one. Question: Can you ﬁnd the�area using the limits of integration 0 and π? π 1 Answer: Yes. The integral (2a cos θ)2 dθ gives a correct answer. 0 2 However, r = 2a cos θ, 0 ≤ θ ≤ π is an awkward way to describe a circle. As θ ranges from 0 to π2 , r is positive and (r, θ) moves along the top half of the circle. As θ sweeps through the second quadrant ( π2 < θ < π), r is negative and so the curve appears in the fourth quadrant. When we work with negative values of r it’s easy to get confused, so when possible it’s a good idea to choose our limits of integration so that r is positive.