N2021 P2 Q2

The diagram shows a sketch of the curve \(y=\mathrm{f}\left( x \right)\). The region under the curve between \(x=1\) and \(x=5\), shown shaded in the diagram, is \(A\). This region is split into \(5\) vertical strips of equal width, \(h\).

1. State the value of \(h\) and show, using a sketch, that \(\sum\limits_{n=0}^{4}{\left( \mathrm{f}\left( 1+nh \right) \right)}\,h\) is less than the area of \(A\).[2]
2. Find a similar expression that is greater than the area of \(A\).[1]

You are now given that \(\mathrm{f}\left( x \right)=\frac{1}{20}{{x}^{2}}+1\).

3. Use the expression given in part (a) and your expression from part (b) to find lower and upper bounds for the area of \(A\).

   [2]
4. Sketch the graph of a function \(y=\mathrm{g}\left( x \right)\), between \(x=1\) and \(x=5\), for which the area between the curve, the \(x\)-axis and the lines \(x=1\) and \(x=5\) is less than \(\sum\limits_{n=0}^{4}{\left( \mathrm{g}\left( 1+nh \right) \right)}\,h\).[1]