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1.8 Integration

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Question 188

A chemical reaction produces a foam whose volume VVV (in cm3\text{cm}^3cm3) changes at a rate V′(t)V'(t)V′(t) where t t\,t is the time in seconds since the start of observation, for t>0t > 0t>0.

Given that

  • V′(t)=(t−9)2ttV'(t) = \dfrac{(t - 9)^2}{t\sqrt{t}}V′(t)=tt​(t−9)2​
  • the observation point P(4,10)P(4, 10)P(4,10) lies on the curve V=g(t)V = g(t)V=g(t)
a.

(i) Find the value of the gradient of the curve at PPP.

(ii) Hence find the equation of the tangent to the curve at PPP, giving your answer in the form at+bV+c=0at + bV + c = 0at+bV+c=0, where aaa, b b\,b and c c\,c are integers to be found.

[4]
b.

Find g(t)g(t)g(t), simplifying your answer.

[5]
Markscheme

1.8 Integration Questions

  1. A Level
  2. /Maths
  3. /1.8 Integration

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.

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