| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 4 |
| Topic | Indefinite & Definite Integrals |
| Type | Limit of sum as integral |
| Difficulty | Standard +0.3 This is a straightforward application of recognizing a Riemann sum and converting it to a definite integral, followed by routine integration using the reverse chain rule. The integration of (1+2x)^(1/2) is a standard textbook exercise requiring only substitution or pattern recognition, making this slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08g Integration as limit of sum: Riemann sums |
2.
\section*{In this question you must show all stages of your working.}
\section*{Solutions relying entirely on calculator technology are not acceptable.}
\begin{enumerate}[label=(\alph*)]
\item Express as an integral
$$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x$$
\item Using your answer to part (a) show that
$$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x = \frac { 98 } { 3 }$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q2 [4]}}