Let \( R \) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \( R \) is revolved about the \( y \)-axis. \( y=5-x, y=0, x=1 \), and \( x=4 \) Set up the integral that gives the volume of the solid using the shell method. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. \( \int_{1}^{4}(2 \pi x(5-x)) d x \) B. \( \int_{1}(\square) \) The volume is (Type an exact answer.)

(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 \( 6 x+a \), where \( a \) 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 ... \( \begin{array}{llll}\text { a) } y=3 x^{2}-4 & \text { b) } y=3 x^{2}-11 x+7 & \text { c) } y=3 x^{2}-4 x & \text { d) } y=3 x^{2}-x-3\end{array} \)

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

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

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 ... \( \begin{array}{llll}\text { a) } y=3 x^{2}-4 & \text { b) } y=3 x^{2}-11 x+7 & \text { c) } y=3 x^{2}-4 x & \text { d) } y=3 x^{2}-x-3\end{array} \)