1.06j

1 The curve \(C\) has equation \(y = \frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\). Show that the length of the \(\operatorname { arc }\) of \(C\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 2 }\) is \(\frac { \mathrm { e } ^ { 2 } - 1 } { 4 \mathrm { e } }\).

7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x\). Show that, for all positive integers \(n\), $$I _ { n } = n I _ { n - 1 } - 1$$ Find the exact value of \(I _ { 4 }\). By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }\) in the interval \(0 \leqslant x \leqslant 1\), show that $$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$

7 The curve \(C\) has equation \(y = \mathrm { e } ^ { - 2 x }\). Find, giving your answers correct to 3 significant figures,

1. the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) over the interval \(0 \leqslant x \leqslant 2\),
2. the coordinates of the centroid of the region bounded by \(C\), \(x = 0\), \(x = 2\) and \(y = 0\).