The profile of a specialized aerodynamic sail is modeled by a curve with parametric equations
x=36−6t,y=t336+6t,0≤t≤6 x = \sqrt{36-6t}, \quad y = \frac{t^3}{\sqrt{36+6t}}, \quad 0 \le t \le 6 x=36−6t,y=36+6tt3,0≤t≤6The curve intersects the yyy-axis at the point where t=6t=6t=6 and the xxx-axis at the point where t=0t=0t=0. The region RRR is bounded by the curve and the positive xxx and yyy axes.
Show that the area of RRR is given by
K∫06t31296−36t2 dt K \int_{0}^{6} \frac{t^3}{\sqrt{1296-36t^2}} \, dt K∫061296−36t2t3dtwhere KKK is a constant to be found.
Using the substitution u=1296−36t2u = 1296 - 36t^2u=1296−36t2, or otherwise, find the exact area of RRR.
