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

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

• 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

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

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

• STEP 1: Find the intersections of the line and the curve
• STEP 2: Find the area under a curve, R_C, using definite integration
• STEP 3: Find the area under a line, R_L, 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_C – R_L
  • If line on top this will be R_L - R_C

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

• 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