Computer Science applicants should turn to page 14. Let $$I ( c ) = \int _ { 0 } ^ { 1 } \left( ( x - c ) ^ { 2 } + c ^ { 2 } \right) \mathrm { d } x$$ where $c$ is a real number. (i) Sketch $y = ( x - 1 ) ^ { 2 } + 1$ for the values $- 1 \leqslant x \leqslant 3$ on the axes below and show on your graph the area represented by the integral $I$ (1). (ii) Without explicitly calculating $I ( c )$, explain why $I ( c ) \geqslant 0$ for any value of $c$. (iii) Calculate $I ( c )$. (iv) What is the minimum value of $I ( c )$ (as $c$ varies)? (v) What is the maximum value of $I ( \sin \theta )$ as $\theta$ varies? [Figure]
(i) [3 marks]
\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science applicants should turn to page 14.\\
Let
$$I ( c ) = \int _ { 0 } ^ { 1 } \left( ( x - c ) ^ { 2 } + c ^ { 2 } \right) \mathrm { d } x$$
where $c$ is a real number.\\
(i) Sketch $y = ( x - 1 ) ^ { 2 } + 1$ for the values $- 1 \leqslant x \leqslant 3$ on the axes below and show on your graph the area represented by the integral $I$ (1).\\
(ii) Without explicitly calculating $I ( c )$, explain why $I ( c ) \geqslant 0$ for any value of $c$.\\
(iii) Calculate $I ( c )$.\\
(iv) What is the minimum value of $I ( c )$ (as $c$ varies)?\\
(v) What is the maximum value of $I ( \sin \theta )$ as $\theta$ varies?\\
\includegraphics[max width=\textwidth, alt={}, center]{88ab9917-8b40-4dbf-a806-1a546164dcee-10_1205_1162_1455_495}