Challenging +1.8 This AEA question requires geometric insight about areas, solving a non-standard equation involving integrals (finding where areas are equal rather than curves intersect), and then generalizing the pattern through a transformation. Part (d) demands creative thinking to construct a function satisfying the given property, going beyond routine integration techniques. However, the individual steps are manageable with clear guidance from earlier parts.
2.(a)On the same diagram,sketch \(y = x\) and \(y = \sqrt { } x\) ,for \(x \geq 0\) ,and mark clearly the coordinates of the points of intersection of the two graphs.
(b)With reference to your sketch,explain why there exists a value \(a\) of \(x ( a > 1 )\) such that
$$\int _ { 0 } ^ { a } x \mathrm {~d} x = \int _ { 0 } ^ { a } \sqrt { } x \mathrm {~d} x$$
(c)Find the exact value of \(a\) .
(d)Hence,or otherwise,find a non-constant function \(\mathrm { f } ( x )\) and a constant \(b ( b \neq 0 )\) such that
$$\int _ { - b } ^ { b } \mathrm { f } ( x ) \mathrm { d } x = \int _ { - b } ^ { b } \sqrt { } [ \mathrm { f } ( x ) ] \mathrm { d } x$$
2.(a)On the same diagram,sketch $y = x$ and $y = \sqrt { } x$ ,for $x \geq 0$ ,and mark clearly the coordinates of the points of intersection of the two graphs.\\
(b)With reference to your sketch,explain why there exists a value $a$ of $x ( a > 1 )$ such that
$$\int _ { 0 } ^ { a } x \mathrm {~d} x = \int _ { 0 } ^ { a } \sqrt { } x \mathrm {~d} x$$
(c)Find the exact value of $a$ .\\
(d)Hence,or otherwise,find a non-constant function $\mathrm { f } ( x )$ and a constant $b ( b \neq 0 )$ such that
$$\int _ { - b } ^ { b } \mathrm { f } ( x ) \mathrm { d } x = \int _ { - b } ^ { b } \sqrt { } [ \mathrm { f } ( x ) ] \mathrm { d } x$$
\hfill \mbox{\textit{Edexcel AEA 2007 Q2 [10]}}