๐Ÿ“š

All Subjects

ย >ย 

โ™พ๏ธย 

AP Calc

ย >ย 

๐Ÿถ

Unit 8

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

2 min readโ€ขjune 8, 2020

anusha

Anusha Tekumulla


AP Calculus AB/BCย โ™พ๏ธ

Bookmarkedย 7.1kย โ€ขย 259ย resources
See Units

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.

Was this guide helpful?

Join us on Discord
Thousands of students are studying with us for the AP Calculus AB/BC exam.
join now
Hours Logo
Studying with Hours = the ultimate focus mode
Start a free study session
๐Ÿ” Are you ready for college apps?
Take this quiz and find out!
Start Quiz
Browse Study Guides By Unit
๐Ÿ“†Big Reviews: Finals & Exam Prep
โœ๏ธFree Response Questions (FRQ)
๐ŸงMultiple Choice Questions (MCQ)
โ™พUnit 10: Infinite Sequences and Series (BC Only)
๐Ÿ‘‘Unit 1: Limits & Continuity
๐Ÿค“Unit 2: Differentiation: Definition & Fundamental Properties
๐Ÿค™๐ŸฝUnit 3: Differentiation: Composite, Implicit & Inverse Functions
๐Ÿ‘€Unit 4: Contextual Applications of the Differentiation
โœจUnit 5: Analytical Applications of Differentiation
๐Ÿ”ฅUnit 6: Integration and Accumulation of Change
๐Ÿ’ŽUnit 7: Differential Equations
๐ŸถUnit 8: Applications of Integration
๐Ÿฆ–Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
FREE AP calc Survival Pack + Cram Chart PDF
Sign up now for instant access to 2 amazing downloads to help you get a 5
Join us on Discord
Thousands of students are studying with us for the AP Calculus AB/BC exam.
join now
๐Ÿ’ช๐Ÿฝ Are you ready for the Calc AB exam?
Take this quiz for a progress check on what youโ€™ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
๐Ÿ’ช๐Ÿฝ Are you ready for the Calc BC exam?
Take this quiz for a progress check on what youโ€™ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
Play this on HyperTyper
Practice your typing skills while reading Finding the Area Between Curves That Intersect at More Than Two Points
Start Game
๐Ÿ“ฑ Stressed or struggling and need to talk to someone?
Talk to a trained counselor for free. It's 100% anonymous.
Text FIVEABLE to 741741 to get started.
ยฉ 2021 Fiveable, Inc.