πŸ“š

Β >Β 

♾️ 

Β >Β 

🐢

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

2 min readβ€’june 8, 2020

Anusha Tekumulla

Anusha Tekumulla


AP Calculus AB/BC ♾️

279Β resources
See Units

When Curves Intersect More Than Once

This topic is very similar to the previous two topics. However,Β  this topic involves more steps. Normally, you’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:Β 
  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)Β 
  3. Set the two equations equal to each other and find the intersection pointsΒ 
  4. Integrate from the intersection points found in step 3 to find the area
If you’re still confused, take a look at this example:Β 

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

Let’s say we want to find the area between y = sin(x) and y = -sin(x) from 0 to 2Ο€.
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.Β 
  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.Β 
  3. Third, we set the functions equal to each other to find the intersection point. Now, we know the intersection point is x = Ο€.Β 
  4. Fourth, we integrate (sin(x) - -sin(x)) from 0 to Ο€ to find the first area. Lastly, we integrate (-sin(x) - sin(x)) from Ο€ to 2Ο€ to find the second area.Β 
  5. Add them together to get the final area.
Browse Study Guides By Unit
πŸ‘‘Unit 1 – Limits & Continuity
πŸ€“Unit 2 – Fundamentals of Differentiation
πŸ€™πŸ½Unit 3 – Composite, Implicit, & Inverse Functions
πŸ‘€Unit 4 – Contextual Applications of Differentiation
✨Unit 5 – Analytical Applications of Differentiation
πŸ”₯Unit 6 – Integration & Accumulation of Change
πŸ’ŽUnit 7 – Differential Equations
🐢Unit 8 – Applications of Integration
πŸ¦–Unit 9 – Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC Only)
β™ΎUnit 10 – Infinite Sequences & Series (BC Only)
🧐Multiple Choice Questions (MCQ)
✍️Free Response Questions (FRQ)
πŸ“†Big Reviews: Finals & Exam Prep

Fiveable
Fiveable
Home
Stay Connected

Β© 2023 Fiveable Inc. All rights reserved.


Β© 2023 Fiveable Inc. All rights reserved.