Let \( R \) be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when \( R \) is revolved about the \( x \)-axis. \( y=4 x^{2} \) and \( y=20-x^{2} \) The volume of the solid is (Type an exact answer.)

Determine the indefinite integral \( \int (6x^3 + 4x^2) \, dx \)

3. \( \lim _{x \rightarrow \infty} \frac{x-1}{2 x+5} \)

Show that \( \Gamma\left(\frac{1}{2}\right)=\sqrt{\pi} \)

(a) Determine whether \( f(x)=\frac{|x-4|-2}{x} \) is continuous at \( x=4 \). [6 marks] (b) Given \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \). By finding the limit at appropriate values, obtain the horizontal and vertical asymptotes, if they exist. Hence, sketch the graphs of \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \).

If the slope of the tangent to the curve: \( y=f(x) \) at any point on it equals \( 6 x+a \), whe \( a \) is constant if the equation of the tangent to the curve at the point \( (1,-1) \) is \( y=4- \) \( 5 x \), then the equation of the curve is ... \( \begin{array}{llll}\text { a) } y=3 x^{2}-4 & \text { b) } y=3 x^{2}-11 x+7 & \text { c) } y=3 x^{2}-4 x & \text { d) } y=3 x^{2}-x-3\end{array} \)