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Textbook by: Serhat Beyenir. E-mail the author

# Problem Set

Module by: Serhat Beyenir. E-mail the author

Summary: Problem Set for Numerical Integration with MATLAB

## Exercise 1

Let the function y defined by y=cosx y x . Plot this function over the interval [-pi,pi]. Use numerical integration techniques to estimate the integral of y over [0, pi/2] using increments of pi/10 and pi/1000.

## Exercise 2

Let the function y defined by y=0.04x22.13x+32.58 y 0.04 x 2 2.13 x 32.58 . Plot this function over the interval [3,30]. Use numerical integration techniques to estimate the integral of y over [3,30].

## Exercise 3

A 2000-liter tank is full of lube oil. It is known that if lube oil is drained from the tank, the mass flow rate will decrease from the maximum when the tank level is at the highest. The following data were collected when the tank was drained.

Table 1: Data
Time [min] Mass Flow [kg/min]
0 50.00
5 48.25
10 46.00
15 42.50
20 37.50
25 30.50
30 19.00
35 9.00

Write a script to estimate the amount of oil drained in 35 minutes.

## Exercise 4

A gas is expanded in an engine cylinder, following the law PV1.3=c. The initial pressure is 2550 kPa and the final pressure is 210 kPa. If the volume at the end of expansion is 0.75 m3, compute the work done by the gas. 1

## Exercise 5

A force F acting on a body at a distance s from a fixed point is given by F=3s+1s2 F 3 s 1 s 2 . Write a script to compute the work done when the body moves from the position where s=1 to that where s=10. 2

## Exercise 6

The pressure p and volume v of a given mass of gas are connected by the relation (p+av2)(vb)=k p a v 2 v b k where a, b and k are constants. Express p in terms of v, and write a script to compute the work done by the gas in expanding from an initial volume to a final volume. 3

Test your solution with the following input:
a: 0.01
b: 0.001
The initial pressure [kPa]: 100
The initial volume [m3]: 1
The final volume [m3]: 2

## Footnotes

1. Applied Heat for Engineers by W. Embleton and L Jackson, Thomas Reed Publications. © 1999, (p. 80)
2. O. N. Mathematics: 2 by J. Dobinson, Penguin Library of Technology. © 1969, (p. 213)
3. O. N. Mathematics: 2 by J. Dobinson, Penguin Library of Technology. © 1969, (p. 212)

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