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,−8)A(3, -8)A(3,−8) 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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.