A robotic welding arm follows a path C C\,C in a horizontal workspace, defined by the parametric equations
x=12t2+2,y=2t−8t,t>0 x = \frac{1}{2}t^2 + 2, \quad y = 2t - \frac{8}{t}, \quad t > 0 x=21t2+2,y=2t−t8,t>0where x x\,x and y y\,y are coordinates in centimetres. The path C C\,C intersects the xxx-axis at the point QQQ.
Determine the xxx-coordinate of QQQ.
A safety barrier is represented by the line lll, which is the normal to the path C C\,C at the point PPP. Given that t=4t = 4t=4 at PPP:
Write down the coordinates of PPP.
Using calculus, show that an equation of l l\,l is
8x+5y=110 8x + 5y = 110 8x+5y=110The region R R\,R is bounded by the path C C\,C from Q Q\,Q to PPP, the line l l\,l from P P\,P to the xxx-axis, and the xxx-axis between Q Q\,Q and the line intercept.
Using algebraic integration, find the exact volume of the solid of revolution formed when the region R R\,R is rotated through 2π 2\pi\,2π radians about the xxx-axis.