The cross-section of a custom-designed drainage channel is modeled by the parametric equations
x=2u2+2u,y=10u(7−u),a≤u≤b x = 2u^2 + 2u, \quad y = \dfrac{10}{u(7 - u)}, \quad a \le u \le b x=2u2+2u,y=u(7−u)10,a≤u≤bwhere a a\,a and b b\,b are constants. The surface of the water in the channel is represented by the horizontal line y=1y = 1y=1, which connects the two endpoints of the curve.
Determine the values of the constants a a\,a and bbb, where b>ab > ab>a.
The region R R\,R represents the area of the cross-section of the water, bounded by the curve and the line y=1y = 1y=1. Show that the area of R R\,R is given by
M−k∫ab2u+1u(7−u) du M - k \int_{a}^{b} \frac{2u+1}{u(7-u)} \, du M−k∫abu(7−u)2u+1duwhere M M\,M and k k\,k are constants to be determined.
Express 2u+1u(7−u)\dfrac{2u+1}{u(7-u)}u(7−u)2u+1 in partial fractions.
Use algebraic integration to find the exact area of RRR, giving your answer in the form A−Bln(2.5)A - B \ln(2.5)A−Bln(2.5), where A A\,A and B B\,B are constants.
