The profile of a high-precision cooling fin for a microchip is modeled by a curve with parametric equations
x=36−6τ,y=τ336+6τ,0≤τ≤6 x = \sqrt{36 - 6\tau}, \quad y = \dfrac{\tau^3}{\sqrt{36 + 6\tau}}, \quad 0 \le \tau \le 6 x=36−6τ,y=36+6ττ3,0≤τ≤6A cross-section of the fin, region RRR, is bounded by this curve, the xxx-axis, and the yyy-axis. The curve touches the xxx-axis at τ=0\tau = 0τ=0 and meets the yyy-axis at τ=6\tau = 6τ=6.
Show that the area of R R\,R is given by
K∫06τ31296−36τ2 dτ K \int_{0}^{6} \dfrac{\tau^3}{\sqrt{1296 - 36\tau^2}} \, d\tau K∫061296−36τ2τ3dτwhere K K\,K is a constant to be found.
Using the substitution u=1296−36τ2u = 1296 - 36\tau^2u=1296−36τ2, or otherwise, determine the exact area of RRR.
