The cross-section of a curved acoustic reflector is modelled by the curve C C\,C with parametric equations
x=−4cos2θ,h=5cosθ,0≤θ≤π2x = -4 \cos 2\theta, \quad h = 5 \cos \theta, \quad 0 \le \theta \le \frac{\pi}{2}x=−4cos2θ,h=5cosθ,0≤θ≤2π
where x x\,x is the horizontal displacement in metres from a central axis and h h\,h is the height in metres. The region R R\,R is bounded by the curve CCC, the xxx-axis, and the hhh-axis for the portion of the curve where x≥0x \ge 0x≥0.
(i) Show, making your working clear, that the area of R=∫π/4π/280cos2θsinθ dθR = \int_{\pi/4}^{\pi/2} 80 \cos^2 \theta \sin \theta \, d\thetaR=∫π/4π/280cos2θsinθdθ.
(ii) Hence find, by algebraic integration, the exact value of the area of RRR.
Show that all points on C C\,C satisfy h=ax+bh = \sqrt{ax+b}h=ax+b, where a a\,a and b b\,b are constants to be found.
State the range of the function f(x)=ax+bf(x) = \sqrt{ax+b}f(x)=ax+b for the domain −4≤x≤4-4 \le x \le 4−4≤x≤4.