Area Between Curve & y-axis (DP IB Analysis & Approaches (AA): HL): Revision Note

Area 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 graph of$y=f\left(x\right)$

  • the$y$-axis

  • the horizontal line$y=a$

  • the horizontal line$y=b$

• 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

$A={\int }_a^b\left|x\right|\mathrm{d}y$

• The function is normally given in the form$y=f\left(x\right)$

  • so will need rearranging into the form$x=g\left(y\right)$

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

  • You can use a GDC to plot the curve and find roots as necessary

• STEP 1
  Identify the limits $a$ and $b$
  Sketch the graph of $y=f\left(x\right)$ or use a GDC to do so, especially if $a$ and $b$ are not given directly in the question
   

• STEP 2
  Rearrange $y=f\left(x\right)$ into the form $x=g\left(y\right)$
  This is similar to finding the inverse function $f^{-1}\left(x\right)$
   

• STEP 3
  Evaluate the formula to evaluate the integral and find the area required
  If using a GDC remember to include the modulus ( | … | ) symbols around $x$ 

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 $\mathit{y}$-axis (negative $x$ values)

  • $\left|x\right|$ makes such areas 'positive' by reflecting them in the $y$-axis

    • Your GDC will apply $\left|x\right|$ automatically as long as you put the modulus symbol ( | ... | ) around $g\left(y\right)$

• Otherwise, to apply ‘$\left|x\right|$’ manually, split the integral into positive and negative parts

  • Write an integral and evaluate each part separately

  • Change any negative values found to positive

  • Then add all the positive values together to give the total area