What can you say about the area under each function on the two cards below, between \(x=1\) and \(x=2\)?

Which is greater?

(a) \[f(x)=\left(4-x^2\right)^{\frac{1}{2}}\]


f of x is 2 over square root 1 plus bracket x minus 1.5 close bracket squared.

Given that we have a graphical representation for the other function, it would make sense to think about what this might be before we dive into any calculations.

We are dealing with the function \(f(x) = (4 - x^2)^{\frac{1}{2}}\). If this were written in the form \(y=(4 - x^2)^{\frac{1}{2}}\) we might be tempted to square both sides of the equation and rearrange to obtain \(x^2 + y^2 = 4,\) which should look familiar!

However, when we produce a sketch we must not lose sight of the original function.

Plot of f of x = square root of 4 minus x squared.
The plot of f is not the full circle because that is not a function.
  • Why is the graphical representation on the left correct? Why is the one on the right incorrect?

  • Where does the function meet the \(x\)-axis?

  • Where does the function intercept the \(y\)-axis?

  • How can we now calculate the area under the function between \(x=1\) and \(x=2\)?

This curve looks as though it it symmetrical about \(x=1.5\). How can we decide for certain if this is the case?

We might notice that this function is a horizontal translation of \(g(x) = \dfrac{2}{\sqrt{1 + x^2}}\).

  • Is this an odd or even function?

  • What can this tell us about the properties of its graphical representation?

  • Why might it be useful for us to know that the original curve is symmetrical about \(x=1.5\)?

  • Can we establish maximum and minimum values for the shaded area that will enable us to determine whether it is greater than or less than the area on card (a)?

What do you notice if you plot the two functions on the same axes?