Find  \(p\) and \(q\),   if  \(\int {\dfrac{{{x^2} + 4x}}{{\sqrt[3]{{{x^3} + 6{x^2}}}}}} dx = \dfrac{1}{p}\sqrt[3]{{{{\left( {{x^3} + 6{x^2}} \right)}^q}}} + C\).

Find \(a\),   if  \(\int_0^3 {x\sqrt {x + 1} } dx = \dfrac{a}{{15}}\).

Find the upper area enclosed by  \(f\left( x \right) = x\sqrt x \),   \(x\)-axis and the vertical lines  \(x = 0\)  and  \(x = 1\)   using 4 intervals, correcting to four significant figures.

Find the value of  \(k\)   if  \(\int_2^k {\dfrac{3}{{{x^2}}}dx} = \dfrac{9}{{10}}\).

Find \(a\),   if  \(\int_{ – 1}^1 {{{\left( {\dfrac{{ – x + 1}}{2}} \right)}^5}} dx = \dfrac{a}{3}\).

Find the area between the  \(x\)-axis  and  \(y = x – 2\)   from  \(x = 1\) to \(x = 3\).

Find the area of the region bounded by   \(y = {x^3} – 2x\) and \(y = 2x\),   correcting to one decimal place.

A particle is initially at the origin, and is stationary. It accelerates according to  \(a\left( t \right) = \dfrac{1}{{{{\left( {t + 1} \right)}^2}}}\)  m/s².   Find the best description of the motion of the particle at  \(t = 3\).

Find the volume of the solid formed when  \(y = {x^3} \)  for  \(0 \le x \le 2\)  is revolved through  \(2\pi \) or \(360^\circ \) about the \(y\)-axis.   correcting to three significant figures.

If the area between the line  \(y = {x^3}\) and the curve  \(y = {x^2}\) is rotated about the  \(x\)-axis, the volume is  \(\dfrac{a}{b}\pi \).   Find \(a\) and \(b\),   where \(a\) and \(b\)  have no common factors.