Computing the area under a curve

Given \mathrm{Area}=\int_2^5 x^2\,d x , an analytical solution would produce 39. Use trapz command and solve it

1. Initialize variable x as a row vector, from 2 with increments of 0.1 to 5: x=2:.1:5;
2. Declare variable y as y=x^2; . Note the following error prompt: ??? Error using ==>mpower Inputs must be a scalar and a square matrix. This is because x is a vector quantity and MATLAB is expecting a scalar input for y. Because of that, we need to compute y as a vector and to do that we will use the dot operator as follows: y=x.^2; . This tells MATLAB to create vector y by taking each x value and raising its power to 2.
3. Now we can issue the following command to calculate the first area, the output will be as follows:

Notice that this numerical value is slightly off. So let us increase the number of increments and calculate the area again:

Yet another increase in the number of increments:

Determine the value of the following integral:

\int_0^{\pi } \sin x\,d x

1. Initialize variable x as a row vector, from 0 with increments of pi/100 to pi: x=0:pi/100:pi;
2. Declare variable y as y=sin(x);
3. Issue the following command to calculate the first area, the output will be as follows:

let us increase the increments as above:

A gas expands according to the law, PV ^1.4 =c. Initially, the pressure is 100 kPa when the volume is 1 m ³ . Write a script to compute the work done by the gas in expanding to three times its original volume O. N. Mathematics: 2 by J. Dobinson, Penguin Library of Technology. © 1969, (p. 184) .

Recall that PV diagrams can be used to estimate the net work performed by a thermodynamic cycle, see Wikipedia or we can use definite integral to compute the work done (WD) as follows:

If we rearrange the expression pressure as a function of volume, we get:

By considering the initial state, we can determine the value of c:

From the equation and the equation above, we can write:

By inserting P in WD , we get:

For MATLAB solution, we will consider P as a function of V and WD . Now, let us apply the three-step approach we have used earlier:

1. Initialize variable volume as a row vector, from 1 with increments of 0.001 to 3: v=1:0.001:3;
2. Declare variable pressure as p=100./v.^1.4;
3. Use the trapz function to calculate the work done, the output will be as follows:

These steps can be combined in an m-file as follows:

A body moves from rest under the action of a direct force given by $F=\frac{15}{x+3}$ where x is the distance in meters from the starting point. Write a script to compute the total work done in moving a distance 10 m. O. N. Mathematics: 2 by J. Dobinson, Penguin Library of Technology. © 1969, (p. 183)

Recall that the general definition of mechanical work is given by the following integral, see Wikipedia :

Therefore we can write:

Applying the steps we followed in the previous examples, we write:

The output of the above code is: