The diagram shows a sketch of the curve $y=\mathrm{f}\left( x \right)$. The region under the curve b

Question

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, $h$. 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] Find a similar expression that is greater than the area of $A$. [1] You are now given that $\mathrm{f}\left( x \right)=\frac{1}{20}{{x}^{2}}+1$. 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] 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]

Accepted Answer

(c) $5.61<A<6.57$

N2021 P2 Q2

N2021 P2 Q2

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