Is Simpson or Trapezium Rule Better for Calculating Area Under a Curve?

[Link]

Why should I use the simpson or trapezium rule when calculating the area under a curve? It is much easier and accurate when using integration the ordinary way

 

[Justin Lazear]

In general, you must approximate something that is either difficult or impossible to do analytically. Not many functions can be "integrated the ordinary way." Most functions can be approximated, though.

--J

 

[Link]

Can you give me an example of a function that is impossible to integrate analytiacally?

 

[Justin Lazear]

$$\mbox{Si}(z) \equiv \int_0^z \frac{\sin{t}}{t} dt$$

--J

 

[HallsofIvy]

$$\int_0^1 e^{x^2}dx$$

"Almost all" integrals cannot be done analytically.

 

[arildno]

However, "almost all" integrals you learn about in your first year can be solved by the "ordinary" way..

 

[scholzie]

Perhaps more tangable to you in the near future: If you are learning Simson's rule now, you will most likely get to arclength very shortly. You will also find then that sometimes evaluating an integral like

$$\int_a^b\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}{dx}$$

Can be very difficult if $\frac{dy}{dx}$ is long or confusing.