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IB DP Maths: AI HL复习笔记5.4.3 Further Applications of Integration

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月22日 下午12:07
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Negative Integrals

  • The area under a curve may appear fully or partially under the x-axisthis occurs if the negative area(s) is/are greater than the positive area(s), their sum will be negative

How do I find the area under a curve when the curve is fully under the x-axis?

STEP 1
Write the expression for the definite integral to find the area as usual
This may involve finding the lower and upper limits from a graph sketch or GDC and f(x) may need to be rewritten in an integrable form
STEP 2
The answer to the definite integral will be negative
Area must always be positive so take the modulus (absolute value) of it

How do I find the area under a curve when all, or some, of the curve is below the x-axis?

Exam Tip

  • If no diagram is provided, quickly sketch one so that you can see where the curve is above and below the x - axis and split up your integrals accordingly
    • You should use your GDC to do this

Worked ExampleQYldKmqS_5-4-3-ib-hl-ai-aa-extraaa-ai-we2-soltnArea Between Curve & y-axis

What is meant by the area between a curve and the y-axis?

  • The area referred to is the region bounded by
  • The exact area can be found by evaluating a definite integral

How do I find the area between a curve and the y-axis?

  • Use the formulaExam Tip
  • If no diagram is provided, quickly sketch one so that you can see where the curve is to the left and right of the y - axis and split up your integrals accordingly
    • You should use your GDC to do this

Worked Example5-4-3-ib-hl-ai-aa-extraaa-ai-we1-soltn

Area Between a Curve and a Line

  • Areas whose boundaries include a curve and a (non-vertical) straight line can be found using integration
    • For an area under a curve a definite integral will be needed
    • For an area under a line the shape formed will be a trapezium or triangle
      • basic area formulae can be used rather than a definite integral
      • using a GDC, one method is not particularly trickier than the other
  • The total area required could be the sum or difference of the area under the curve and the area under the line

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

STEP 1
If a diagram is not given, use a GDC to draw the graphs of the curve and line and identify the area to be found
STEP 2
Use a GDC to find the root(s) of the curve, the root of the line and the