AQA A Level Maths: Pure复习笔记8.1.5 Area between a curve and a line

Area between a curve and a line

What is the area between a curve and a straight line?

8.1.5-Notes-fig1

  • This is the area enclosed by a curve and a line between their intersections
  • The intersections may need to be found using simultaneous equations

How do I find the area between a curve and a line?

  • The area enclosed will be the difference between …
    • the area under the curve and
    • the area under the line
  • These can be found separately

8.1.5-Notes-fig2

  • There may be easier ways to find the area under a line

8.1.5-Notes-fig3

  • Once the areas are found a question can be answered in full

8.1.5-Notes-fig4

8.1.5-Notes-fig5

8.1.5-Notes-fig6

  • STEP 1: Find the intersections of the line and the curve
  • STEP 2: Find the area under a curve, RC, using definite integration
  • STEP 3: Find the area under a line, RL, either using definite integration or the area formulae for basic shapes
  • STEP 4: To find the area, R, between the curve and the line subtract the smaller area from the larger area
    • If curve on top this will be R– RL
    • If line on top this will be RL - RC

Alternative method

  • An alternative method allows subtraction of the two functions first
  • But be extremely careful as to whether the curve or the line is on top

8.1.5-Notes-fig7

  • STEP 1: Find the intersections of the line and the curve
  • STEP 2: Subtract the functions algebraically
    • Ensure this is done the correct way round!
  • STEP 3: Find the area, R, between the curve and the line using definite integration

8.1.5-Notes-fig88.1.5-Notes-fig9

Exam Tip

  • Add information to any diagram provided
  • Add axes intercepts, as well as intercepts between lines and curves
  • Mark and shade the area you’re trying to find
  • If no diagram provided, sketch one!

Worked Example

8.1.5-Example-Question-fig18.1.5-Example-fig28.1.5-Example-fig38.1.5-Example-fig4

 

 

 

 

 

 

 

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