integration example with matlab

Question

Can someonse show me how to calculate the following in matlab:

where

g = 9.91
As = 1000
z = 10
zv = 30
zD = 10
pz = 999
Az = 500

Answer 1

Well, there are a couple of ways to go about this:

Analytical approach

Like Rudy commented, this is a trivial integral to do analytically because it's the integral of a power (z¹). Its indefinite integral would be:

F(z) = (g/As)(z2/2 - zvz)pzAz

According to the Second Fundamental Theorem of Calculus, the definite integral is computed by substituting the integral bounds into F(z), that is:

ST = F(zd) - F(0) = (g/As)(zD2/2 - zvzD)pzAz

In MATLAB, that would be:

ST = g / As * zD * (zD / 2 - zv) * pz * Az
ST = 
      -1.2375e+06

Numerical approach

If, for some reason, you want to calculate that numerically by approximating the integral, you can use the trapz function for this:

dz = 0.001; z = 0:dz:zD;
y = g / As * (z - zv) * pz * Az;
ST = trapz(z, y)
ST = 
      -1.2375e+06

or make an even cruder approximation with a simple sum:

ST = sum(y * dz)
ST =
      -1.2387e+06