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