Parametric arc length calculation

3 The curve \(C\) has parametric equations $$\mathrm { x } = \mathrm { e } ^ { \mathrm { t } } - \frac { 1 } { 3 } \mathrm { t } ^ { 3 } , \quad \mathrm { y } = 4 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { t } } ( \mathrm { t } - 2 ) , \quad \text { for } 0 \leqslant t \leqslant 2$$ Find, in terms of e , the length of \(C\).

5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 2 } { 3 } \mathrm { t } ^ { \frac { 3 } { 2 } } - 2 \mathrm { t } ^ { \frac { 1 } { 2 } } , \quad \mathrm { y } = 2 \mathrm { t } + 5 , \quad \text { for } 0 < t \leqslant 3$$

1. Find the exact length of \(C\).
2. Find the set of values of \(t\) for which \(\frac { d ^ { 2 } y } { d x ^ { 2 } } > 0\).

1. The curve \(C\) has equation

$$y = \frac { x ^ { 2 } } { 8 } - \ln x , \quad 2 \leqslant x \leqslant 3$$ Find the length of the curve \(C\) giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers to be found.

9 The curve \(C\) has parametric equations $$x = 4 t + 2 t ^ { \frac { 3 } { 2 } } , \quad y = 4 t - 2 t ^ { \frac { 3 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 4$$ Find the arc length of \(C\), giving your answer correct to 3 significant figures. Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 32\).

1 The curve \(C\) is defined parametrically by $$x = \mathrm { e } ^ { t } - t , \quad y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } t }$$ Find the length of the arc of \(C\) from the point where \(t = 0\) to the point where \(t = 3\).

1 The curve \(C\) is defined parametrically by $$x = t ^ { 4 } - 4 \ln t , \quad y = 4 t ^ { 2 }$$ Show that the length of the arc of \(C\) from the point where \(t = 2\) to the point where \(t = 4\) is $$240 + 4 \ln 2 .$$

2 A curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } \cos t , \quad y = \mathrm { e } ^ { t } \sin t , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ Find the arc length of \(C\).

6

1. Given that $$x = \ln ( \sec t + \tan t ) - \sin t$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
2. A curve is given parametrically by the equations $$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$ The length of the arc of the curve between the points where \(t = 0\) and \(t = \frac { \pi } { 3 }\) is denoted by \(s\). Show that \(s = \ln p\), where \(p\) is an integer.