We're gunning for the area of this region here: Let's find the area inside the graph *r* = 2cos *θ* and subtract the area inside the graph *r* = cos θ. The area inside *r* = 2cos θ is and the area inside *r* = cos θ is So the area of the region in between the two graphs is Since the limits of integration on the two integrals are the same, we can combine them into the single integral We could also find this area one slice at a time. That is, we could find the area of the region between the graphs *r* = 2cos *θ* and *r* = cos *θ* by slicing the larger region into pizza slices, figuring out the area of the "crust" on each slice, and adding those areas up. The area of the full slice is and the area of the juicy center part is , so the area of the crust is This simplifies to When we add up all all the crust areas and let the number of pieces approach ∞, we get the integral To generalize with a nice, neat, pizza-making formula, when we have a graph *r*_{outer} and a graph *r*_{inner}, the area in between the graphs for α ≤ θ ≤ β is **Be Careful:**
When working with two different radii, we don't want the formula below, for the area between the graphs *r*_{outer} and *r*_{inner}. This may look less complicated, but it's wrong. *The two radii must be squared separately and then subtracted*. |