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BTEC Unit 7 Calculus Assignment: Integration Methods, Motion Analysis, Work & Cooling Law Applications
| University | Milton Keynes College |
| Subject | Unit 7: Calculus to solve engineering problems |
Unit number and title
Unit 7: Calculus to solve engineering problems
Learning aim(s)
B: Examine how Integral calculus can be used to solve engineering problems
Assignment title
Solving engineering problems that involve differentiation
Vocational Scenario or Context
You are working as an apprentice engineer at a company involved in the research, design production and maintenance of bespoke engineering solutions for larger customers.
Part of your apprenticeship is to spend time working in all departments, however a certain level of understanding needs to be shown before the managing director allows apprentices into the design team and so she has developed a series of questions on integration to determine if you are suitable.
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Task 1
1. Integration Problems
The tasks are to:
a) Find the indefinite integral of the function ∫(3x² + 2x + 1)dx
b) Calculate the definite integral ∫14 (3x² + 2x + 1)dx
2. Motion with Uniform Acceleration
An object is moving with a uniform acceleration a, determine the functions for:
a) Velocity – given v = ∫a dt
b) Displacement – given s = ∫v dt
c) Calculate the values of v and s for:
i) a = 2 m/s², initial velocity u = 3 m/s, initial displacement s₀ = 0
ii) a = 9.81 m/s², initial velocity u = 0 m/s, initial displacement s₀ = 10 m
3. Work Done Calculation
The extension, x, of a material with an applied force, F, is given by F = kx.
a) Calculate the work done if the force increases from 100N to 500N using:
i) An analytical integration technique
ii) A numerical integration technique
[Note: the work done is given by the area under the curve]
b) Compare the two answers
c) Using a computer spreadsheet increase the number of values used for your numerical method
d) Analyse any affect the size of numerical step has on the result.
4. Periodic Function Analysis
For the function y = sin(θ), calculate the:
a) Mean
b) Root mean square (RMS)
Over a range of 0 to 2π radians.
[Note the trigonometric identity sin²θ = ½(1 – cos2θ)]
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5. Substitution Method
A complex function can be modelled by the equation: y = 2x(3x² + 4)⁵
Find the indefinite integral of the function using a substitution method.
6. Integration by Parts
The acceleration of an object moving in a strange way has been modelled as a = t²eᵗ.
a) Use integration by parts to find an equation to model the velocity if v = ∫a dt.
b) Is the problem any different if you find ∫t³eᵗ dt?
7. Newton’s Law of Cooling
Newton’s laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation:
dT/dt = -k(T – Troom)
This can also be written as:
dT/(T – Troom) = -k dt
Where:
T = temperature
Troom = room temperature
k = cooling constant
a) Integrate both sides of the equation and show that the temperature difference is given by:
T – Troom = (T0 – Troom)e-kt
b) Calculate T if the initial temperature is 90°C and Troom = 20°C, k = 0.1.
Checklist of evidence required
Your informal report should contain:
- Analysis
- Worked solutions to the problems
Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented.
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Criteria covered by this task:
7/B.D1 Evaluate, using technically correct language and a logical structure, the correct integral calculus and numerical integration solutions for each type of given routine and non-routine functions, including at least two set in an engineering context.
7/B.M2 Find accurately the integral calculus and numerical integration solutions for each type of given routine and non-routine function, and find the properties of periodic functions.
7/B.P4 Find the indefinite integral for each type of given routine function.
7/B.P5 Find the numerical value of the definite integral for each type of given routine function.
7/B.P6 Find, using numerical integration and integral calculus, the area under curves for each type of given routine definitive function.
Sources of information to support you with this Assignment
- mathsisfun.com/index.htm
- mathcentre.ac.uk/students/topics
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