\( \int \int _ { s } f ( x , y , z ) d s = \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 2 \pi } f ( s ( \phi , \theta ) ) \| \frac { \partial s } { \partial \phi } \times \frac { \partial s } { \partial \theta } \| d \theta d \phi \)

\( \int \frac { d x } { x ^ { r } + 4 x + 11 } \)

v) \( \int \frac{x^{3} d x}{\left(x^{2}-a^{2}\right)^{3 / 2}} \)

A particle \( P \) moves in a straight line so that, \( t \) seconds after passing a fixed point \( O \), its displacement, \( s \) metres, is given by \( s=4 t^{3}-15 t^{2}+18 t \). Find (i) the velocity of \( P \) when \( t=3 \), (ii) the acceleration of \( P \) when \( t=3 \), (iii) the values of \( t \) when \( P \) is at instantaneous rest, (iv) the distance travelled by \( P \) in the first 2 seconds of its motion, (v) the distance of \( P \) from \( O \) when \( t=2 \).

derivatives of the following functions (b) \( x^{3} \)

3. Given \( T P=90 \mathrm{~K}^{2}-\mathrm{K}^{3} \) find a. Marginal product b. Average product c. The value of output which equates average product and marginal product