Finding the Area Between Two Curves - Simple Explanation

We'll now explore some special cases when finding areas between curves.In some cases, regions are bounded vertically rather than horizontally.Consider these two curves: f of x in green and g of x in coral.They intersect at these two points, creating a region between them.When the region is better described as being vertically bounded, we need to switch our approach.Instead of integrating with respect to x, we integrate with respect to y. The formula becomes: Area equals the integral from c to d of the absolute value of x-sub-2 of y minus x-sub-1 of y, d y.Here, c and d represent the y-coordinates of the intersection points, and the functions x-1 and x-2 describe the left and right boundaries of the region.Next, let's examine cases where curves intersect multiple times.When we have two curves that intersect more than twice, they create multiple bounded regions.Here, our curves intersect at three points, creating two separate bounded regions.To calculate the total area, we must divide the problem into subregions.For the first region, we integrate j of x minus h of x from a to b.For the second region, we must switch the order since the top and bottom functions have changed positions. We integrate h of x minus j of x from b to c.The total area is the sum of these individual regions.Finally, let's address how to handle cases where we need to ensure we're calculating positive areas.Consider these two simple linear functions that cross at one point.To the left of the intersection, the coral function is above the green function.To the right of the intersection, the green function is above the coral function.To ensure we always calculate a positive area, we can use the absolute value in our integral formula.This is equivalent to integrating q of x minus p of x where q is above p.And integrating p of x minus q of x where p is above q.Let's summarize these special cases for calculating area between curves.Remember to adapt your approach based on how the region is bounded, whether horizontally or vertically. Divide complex regions into simpler subregions. And always ensure you're calculating positive areas by using absolute value or proper function ordering.