The cross-section of a industrial drainage channel is modeled by a curve C C\,C with equation
y=(x−k)2x,x>0 y = \frac{(x - k)^2}{\sqrt{x}}, \quad x > 0 y=x(x−k)2,x>0where k k\,k is a positive constant.
Show that
∫116(x−k)2x dx=ak2+bk+20465 \int_{1}^{16} \frac{(x - k)^2}{\sqrt{x}} \, dx = ak^2 + bk + \frac{2046}{5} ∫116x(x−k)2dx=ak2+bk+52046where a a\,a and b b\,b are integers to be found.
A sketch of the curve C C\,C and a straight line l l\,l are shown. The line l l\,l represents the water level during a flood, intersecting the curve C C\,C at point A(1,9)A(1, 9)A(1,9) and at point B(16,q)B(16, q)B(16,q), where q q\,q is a constant.
Show that k=4k = 4k=4.
The region R R\,R is the cross-sectional area of the water, bounded by the curve C C\,C and the line l l\,l between points A A\,A and BBB. Using the results from parts (a) and (b),
find the area of region RRR.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
