8.6 Finding the Area Between Curves That Intersect at More Than Two Points

When Curves Intersect More Than Once

1. Graph the functions
2. Identify the areas and what approach to take (either top minus bottom with vertical slices or right minus left with horizontal slices) 
3. Set the two equations equal to each other and find the intersection points 
4. Integrate from the intersection points found in step 3 to find the area

Example Problem: Finding the Area Between Two Curves That Intersect at More Than Two Points

1. First, we graph the functions. From this, we find that the functions intersect at more than the endpoints. 
2. We have two areas, and the curves are on the top and bottom for both. Thus, we will use vertical slices to find the areas. 
3. Third, we set the functions equal to each other to find the intersection point. Now, we know the intersection point is x = π. 
4. Fourth, we integrate (sin(x) - -sin(x)) from 0 to π to find the first area. Lastly, we integrate (-sin(x) - sin(x)) from π to 2π to find the second area. 
5. Add them together to get the final area.