9758/2021/P2/Q02

The diagram shows a sketch of the curve $y=f(x)$. 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, $h$.

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(a) State the value of $h$ and show, using a sketch, that $\sum_{n=0}^4(\mathrm{f}(1+n h)) h$ is less than the area of $A$.

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(b) Find a similar expression that is greater than the area of $A$.

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You are now given that $\mathrm{f}(x)=\frac{1}{20} x^2+1$
(c) Use the expression given in part (a) and your expression from part (b) to find lower and upper bounds for the area of $A$.

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(d) Sketch the graph of a function $y=\mathrm{g}(x)$, 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 $\sum_{n=0}^4(\mathrm{~g}(1+n h)) h$.

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