A research probe is submerged in a fluid where the vertical force, F F\,F kilonewtons, exerted on its hull is modeled by the function
F(d)=54d2+4d−26,d>0 F(d) = \frac{54}{d^2} + 4d - 26, \quad d > 0 F(d)=d254+4d−26,d>0where d d\,d is the depth in metres below the surface.
Using calculus,
determine the range of depths for which the vertical force F(d)F(d)F(d) is increasing.
show that ∫39(54d2+4d−26)dd=0\displaystyle \int_{3}^{9} \left( \frac{54}{d^2} + 4d - 26 \right) dd = 0∫39(d254+4d−26)dd=0.
The points A(3,−6)A(3, -6)A(3,−6) and B(6,−0.5)B(6, -0.5)B(6,−0.5) lie on the curve F(d)F(d)F(d).
Given that ∫36(54d2+4d−26)dd=−15\displaystyle \int_{3}^{6} \left( \frac{54}{d^2} + 4d - 26 \right) dd = -15∫36(d254+4d−26)dd=−15.
(i) state the value of ∫69(54d2+4d−26)dd\displaystyle \int_{6}^{9} \left( \frac{54}{d^2} + 4d - 26 \right) dd∫69(d254+4d−26)dd.
(ii) find the value of the constant k k\,k such that ∫36(54d2+4d+k)dd=0\displaystyle \int_{3}^{6} \left( \frac{54}{d^2} + 4d + k \right) dd = 0∫36(d254+4d+k)dd=0.
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.
