2021 ASRJC P2 Q5

It is given that \(R\) is the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x=1\) and the curve \(y={{x}^{2}}+1\).

The area of \(R\) can be estimated by calculating the sum of the areas of trapeziums with equal widths. The diagram below (not drawn to scale) shows an example of two such trapeziums.

[Area of trapezium \(=\frac{1}{2}\times \) sum of parallel sides \(\times \) height]

1. Find the total area of the trapeziums shown in the diagram.[2]

To better estimate the area of \(R\), \(n\) trapeziums of equal width are drawn.

2. State the length of the shorter side of the \({{k}^{\mathrm{th}}}\) trapezium for \(1\le k\le n\). Hence show that its area is given by
   \(\frac{{{\left( k-1 \right)}^{2}}+{{k}^{2}}}{2{{n}^{3}}}+\frac{1}{n}\) units\(^{2}\).[3]

The sum of the areas of the n trapeziums is given as \(A\) units\(^{2}\).

3. Find \(A\), giving your answers in terms of \(n\).[4]
   [You may use \(\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}}\)]
4. State whether \(A\) is an overestimate or an underestimate of the area of region \(R\).[1]
5. Use your answer in part (iii) to find the exact area for region \(R\).[1]