Week 8 Exercises
8.4 Sheet 17 Approximate Integration
Exercise 1
An irregular area has a horizontal base of length 84 mm. The vertical heights measured at regular intervals of 12 mm are, 25.0, 24.5, 23.5, 21.0, 18.0, 14.0, 9.0 mm. Using the midordinate rule, calculate the area of the shape.
Exercise 2
An indicator diagram for a steam engine is 90 mm long. Seven evenly spaced ordinates, including the end ordinates are measured with the following results:- \[ 5.10, 4.60, 3.20, 2.70, 2.32, 2.18, 2.06, \text{ mm} \] Determine the mean pressure in the engine cylinder using the trapezium rule. The pressure scale is 10 kPa to 1 mm. The mean pressure is given by the ratio of the diagram area to the base length.
Exercise 3
By drawing a semi-circle of radius 100 mm, determine the percentage error to two decimal places in the area calculated using i) the mid-ordinate rule ii) the trapezium rule and iii) Simpson’s rule. The ordinate spacing is to be taken as 20 mm.
Exercise 4
A vehicle starts from rest and its velocity is measured every second for sixty seconds, the results being:-
| Velocity (m/s) | Time (s) |
|---|---|
| 0 | 0 |
| 1.2 | 1.0 |
| 2.4 | 2.0 |
| 3.7 | 3.0 |
| 5.2 | 4.0 |
| 6.0 | 5.0 |
| 9.2 | 6.0 |
Determine the distance travelled in the six seconds and the average speed over this time using Simpson’s rule.
Exercise 6
The table below relates the values of y at regular intervals of x
| \(x\) | \(y\) |
|---|---|
| 1.00 | 2.45 |
| 1.25 | 2.80 |
| 1.50 | 3.44 |
| 1.75 | 4.20 |
| 2.00 | 4.33 |
| 2.25 | 3.97 |
| 2.50 | 3.12 |
| 2.75 | 2.38 |
| 3.00 | 1.80 |
Using Simpson’s rule with 8 intervals evaluate \(\displaystyle\int y\ dx\).
Exercise 8
A pin moves along a straight guide such that its velocity \(v\) (\(m.s^{-1}\)) when it is at a distance \(s\) (\(m\)) from the start after time \(t\) (\(s\)) is as given below
| \(v\) (m/s) | \(t\) (s) |
|---|---|
| 0 | 0 |
| 4.00 | 0.50 |
| 7.94 | 1.00 |
| 11.68 | 1.50 |
| 14.97 | 2.00 |
| 17.39 | 2.50 |
| 18.25 | 3.00 |
| 16.08 | 3.50 |
| 0 | 4.00 |
Apply Simpson’s rule with 8 intervals to determine the approximate distance travelled by the pin for the time interval \(t = 0\) to \(t = 4\) s
Exercise 9
Evaluate correct to three decimal places
- \(\displaystyle\int_0^1\sqrt{x}\cos(x)\ dx\),
- \(\displaystyle\int_0^1\sqrt{x}\sin(x)\ dx\).
Exercise 10
By using Simpson’s rule with six intervals, determine the approximate value of \[ \int_0^{\pi/2}\sqrt{(2.5-1.5\cos(2\theta))}\ d\theta. \]
Exercise 11
By using Simpson’s rule with 8 intervals, establish the approximate value of \[ \int_0^1\sqrt{4+x^4}\ dx \]
[Solutions: 1. \(1620\); 2. \(31\,\text{kPa}\); 3. \(0.97\%\), \(3.34\%\), \(1.3\%\); 4. \(3.78\,\text{ms}^{-1}\); 5. \(0.39\); 6. \(6.62\,\text{units}^2\); 7. \(1.19\); 8. \(46.5\,\text{m}\); 9. (i) \(0.529\), (ii) \(0.365\); 10. \(2.41\); 11. \(2.05\);]