The profile of a precision parabolic reflector is modeled by the curve with equation
y=3x2+512x3−100,x>0 y = 3x^2 + \frac{512}{\sqrt{x^3}} - 100, \quad x > 0 y=3x2+x3512−100,x>0where xxx and yyy are measured in decimetres. The point PPP is the only stationary point on the curve.
Use calculus to show that the xxx-coordinate of PPP is 4.
A horizontal support strut, lll, is positioned such that it passes through point PPP and is parallel to the xxx-axis. The region RRR is bounded by the reflector's profile, the strut lll, and the vertical casing at x=1x = 1x=1.
Use algebraic integration to find the exact area of RRR.
