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1.8 Integration

1.8 Integration

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Question 311

A deep-sea research probe is descending through a thermal layer. Its altitude h h\,h in metres, relative to the layer's boundary, is modeled by the function h(t)h(t)h(t) for t>0t > 0t>0, where t t\,t is the time in seconds since the descent began. It is known that:

  • at time t=4t = 4t=4, the probe is at point P(4,−10)P(4, -10)P(4,−10)
  • the rate of change of altitude is given by h′(t)=5t2+at+b3th'(t) = \dfrac{5t^2 + at + b}{3\sqrt{t}}h′(t)=3t​5t2+at+b​, where a a\,a and b b\,b are constants
  • at the instant t=4t = 4t=4, the rate of change of altitude is 12 m s-1
a.

Show that 4a+b=−84a + b = -84a+b=−8.

[3]
b.

Given also that a+2b=−23a + 2b = -23a+2b=−23

Find, in simplest form, the expression for h(t)h(t)h(t).

[6]
c.

The probe's mission data is later adjusted by a time-shift, such that the altitude is modeled by H(t)=h(t−3)H(t) = h(t - 3)H(t)=h(t−3). Given that point P P\,P on the original model is transformed to point Q Q\,Q on the new model,

State the coordinates of QQQ.

[1]
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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