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<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759818780258-bsrboixyhe.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4MTg3ODAyNTgtYnNyYm9peHloZS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzMxNjc3LCJleHAiOjE3OTI3MzE5Nzd9.Xkr1FqvuZ7kdI8WdKVcCVrIMqPgOoPl97bGE32XsFB0" width="400" class="aligncenter" /></figure>
<p>The graph of $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{x}^{2}}+2$, $0\le x\le 2$, is shown. Let $R$ be the region bounded by the $x$-axis, the curve$y={{x}^{2}}+2$, and the lines $x=0$ and $x=2$. Let $n$ rectangles of equal width $\frac{2}{n}$ be used to estimate the area of $R$, as shown in the diagram above.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that $\sum\limits_{i=1}^{n}{{{i}^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}}$, show that the total area of $n$ rectangles is $\frac{4\left( n+1 \right)\left( 2n+1 \right)}{3{{n}^{2}}}+4$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence find the least value of $n$ such that the total area of $n$ rectangles differs from the area of $R$ by less than $0.1$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Instead, $n$ trapeziums of equal width are used to estimate the area of $R$.</p>
<p>For example, when $n=2$, the .trapeziums look like this:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759818823477-2mxh7wcxtyu.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4MTg4MjM0NzctMm14aDd3Y3h0eXUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTY3NywiZXhwIjoxNzkyNzMxOTc3fQ.qG2Ulp6FsG6eJTplMo49eR-YlpU6z9YtlcF8lKSRzXY" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the difference between the total area of the trapeziums and the area of $R$ when $4$ trapeziums are used.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Using your answers to part <strong>(ii)</strong> and part <strong>(iii)</strong>, comment on the use of trapeziums over rectangles to estimate the value of $R$.</div><p style="text-align: right;">[1]</p></li></ol>

The graph of $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{x}^{2}}+2$, $0\le x\le 2$, is shown. Let $R$ be the region bounded by the $x$-axis, the curve$y={{x}^{2}}+2$, and the lines $x=0$ and $x=2$. Let $n$ rectangles of equal width $\frac{2}{n}$ be used to estimate the area of $R$, as shown in the diagram above.

Instead, $n$ trapeziums of equal width are used to estimate the area of $R$.

For example, when $n=2$, the .trapeziums look like this:

Problems involving inequalities

Trapezoidal Rule

Summing Polynomial Series

Riemann Sums

2021 EJC P1 Q8

Applications of Integration

Sequence and Series

<p>It is given that&nbsp;$R$ is the region bounded by the $x$-axis, the $y$-axis, the line $x=1$ and the curve $y={{x}^{2}}+1$.</p>
<p>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.</p>
<p>[Area of trapezium $=\frac{1}{2}\times $ sum of parallel sides $\times $ height]</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759491663803-u89x5nojtmo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk0OTE2NjM4MDMtdTg5eDVub2p0bW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTY3NywiZXhwIjoxNzkyNzMxOTc3fQ.UbduV0uSRP1MBexKxtdqEKssbjFKv9SKdIcNnZb_oDY" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-1759491676501-9ki7d6gp7" data-list-item-path="item-1759491676501-9ki7d6gp7"><div class="list-item-paragraph">Find the total area of the trapeziums shown in the diagram.<span class="inline-marks">[2]</span></div></li></ol>
<p>To better estimate the area of $R$, $n$ trapeziums of equal width are drawn.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759491734432-n7jr72op2" data-list-item-path="item-1759491734432-n7jr72op2"><div class="list-item-content"><span class="list-item-text">State the length of the shorter side of the ${{k}^{\mathrm{th}}}$ trapezium for&nbsp;$1\le k\le n$. Hence show that its area is given by</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{{{\left( k-1 \right)}^{2}}+{{k}^{2}}}{2{{n}^{3}}}+\frac{1}{n}$ units$^{2}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The sum of the areas of the n trapeziums is given as&nbsp;$A$ units$^{2}$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1759491766671-emlbn40ew" data-list-item-path="item-1759491766671-emlbn40ew"><div class="list-item-content"><span class="list-item-text">Find $A$, giving your answers in terms of $n$.</span></div><div class="list-item-paragraph">[You may use $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}}$]<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759491768863-a6w9luolp" data-list-item-path="item-1759491768863-a6w9luolp"><div class="list-item-paragraph">State whether $A$ is an overestimate or an underestimate of the area of region $R$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759491769486-0twpci7ub" data-list-item-path="item-1759491769486-0twpci7ub"><div class="list-item-paragraph">Use your answer in part (iii) to find the exact area for region $R$.<span class="inline-marks">[1]</span></div></li></ol>

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]

Find the total area of the trapeziums shown in the diagram.

To better estimate the area of $R$, $n$ trapeziums of equal width are drawn.

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

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

Find $A$, giving your answers in terms of $n$.

State whether $A$ is an overestimate or an underestimate of the area of region $R$.

Use your answer in part (iii) to find the exact area for region $R$.

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