Area Between Curve & y-axis (DP IB Applications & Interpretation (AI)): Revision Note

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Area between curve & y-axis

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

Graph with curve y=f(x) illustrating region R enclosed by the y-axis, the horizontal line segments y=a and y=b, and the curve. The area of the region is given by the integral of |x|dy from a to b.
  • The area referred to is the region bounded by

    • the graph ofspace y equals f left parenthesis x right parenthesis

    • thespace y-axis

    • the horizontal linespace y equals a

    • the horizontal linespace y equals b

  • 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 formula

space A equals integral subscript a superscript b open vertical bar x close vertical bar space straight d y

  • The function is normally given in the formspace y equals f left parenthesis x right parenthesis

    • so will need rearranging into the formspace x equals g left parenthesis y right parenthesis

  • a and b may not be given directly and could involve the the x-axis (y equals 0) and/or a root ofspace x equals g left parenthesis y right parenthesis (i.e. a point where the function intersects or touches the y-axis)

    • Use a GDC to plot the curve and find roots as necessary

Examiner Tips and Tricks

The integral area formula for area between a curve and the y-axis is given in the exam formula booklet.

  • STEP 1

    If a diagram is not given, use a GDC to draw the graph ofspace y equals f left parenthesis x right parenthesis
    (orspace x equals g left parenthesis y right parenthesis if already given in that form)
    If not identifiable from the question, use the graph to find the limits a and b
     

  • STEP 2

    If needed, rearrangespace y equals f left parenthesis x right parenthesis into the formspace x equals g left parenthesis y right parenthesis

     

  • STEP 3

    Write down the definite integral needed to find the required area

    Use a GDC to evaluate it

    • Your GDC may require to enter the function ‘x’ as the variable (not ‘y’)

    • Remember to include the modulus ( | … | ) symbols around the function

      • Modulus may be called ‘Absolute value (Abs)’ on some GDCs

Examiner Tips and Tricks

If a diagram is not provided, sketching one can really help in this sort of question. Your GDC can help with this.

Can there be 'negative integrals' for areas between a curve and the y-axis?

  • In trickier problems some (or all) of the area may be 'negative'

    • This would be any area that is to the left of the bold italic y-axis (negative x values)

    • open vertical bar x close vertical bar makes such areas 'positive' by reflecting them in the y-axis

      • Your GDC will apply open vertical bar x close vertical bar automatically as long as you put the modulus symbol ( | ... | ) around g open parentheses y close parentheses

Worked Example

Find the area enclosed by the curve with equationspace y equals 2 plus square root of x plus 4 end root, the y-axis and the horizontal lines with equations y equals 3 and y equals 6.

5-4-3-ib-hl-ai-aa-extraaa-ai-we1-soltn

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Paul

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Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

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