| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | December |
| Marks | 4 |
| Topic | Indefinite & Definite Integrals |
| Type | Improper integral evaluation |
| Difficulty | Moderate -0.3 Part (i) is a straightforward integration using the power rule (x^{-4} integrates to -1/(3x^3)), requiring only basic calculus technique. Part (ii) involves an improper integral with a limit as the upper bound approaches infinity, but the calculation is mechanical once the limit is set up correctly. This is slightly easier than average because it's a standard textbook exercise on improper integrals with no conceptual subtlety or problem-solving required beyond direct application of techniques. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
Let $a, b$ satisfy $0 < a < b$.
\begin{enumerate}[label=(\roman*)]
\item Find, in terms of $a$ and $b$, the value of
$$\int_a^b \frac{81}{x^4} dx$$ [2]
\item Explaining clearly any limiting processes used, find the value of $a$, given that
$$\int_a^{\infty} \frac{81}{x^4} dx = \frac{216}{125}$$ [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q2 [4]}}