ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

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

#areabetweencurves

#integration

โฑ๏ธย ย 2 min read

written by

Anusha Tekumulla

anusha tekumulla


When Curves Intersect More Than Once

This topic is very similar to the previous two topics. However,ย  this topic involves more steps. Normally, youโ€™d integrate the function (top minus bottom) or (right minus left) from the endpoints provided. In the situation where the curves intersect at more than two points, you must use additional steps:ย 

  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)ย 

  3. Set the two equations equal to each other and find the intersection pointsย 

  4. Integrate from the intersection points found in step 3 to find the area

If youโ€™re still confused, take a look at this example:ย 

๐Ÿ” Example Problem: Finding the Area Between Two Curves That Intersect at More Than Two Points

Letโ€™s say we want to find the area between y = sin(x) and y = -sin(x) from 0 to 2ฯ€.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(661).png?alt=media&token=296dbba2-7f83-4009-9f33-a087ca4441a8

  1. First, we graph the functions. From this, we find that the functions intersect at more than the endpoints.ย 

  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.ย 

  3. Third, we set the functions equal to each other to find the intersection point. Now, we know the intersection point is x = ฯ€.ย 

  4. Fourth, we integrate (sin(x) - -sin(x)) from 0 to ฯ€ to find the first area. Lastly, we integrate (-sin(x) - sin(x)) from ฯ€ to 2ฯ€ to find the second area.ย 

  5. Add them together to get the final area.

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