Course notes/Numerical integration
Numerical integration is the process of going background from a given derivative of a function to the function itself using computational tools. The simplest form of numerical integration takes advantage of the approximation to the derivative that we get but stopping short of letting $h$ go all the way to zero. That is,
- $f'(x)\approx \frac{f(x+h)-f(x)}{h}$
for a sufficiently small value of $h$. We'll use $h=0.1$ for now.
Suppose we are given a velocity function $v(t)$ but not the position function $x(t)$ and are told that
- $\frac{dx}{dt} = v(t)= e^{-t^2}$.
How do we find the function $x(t)$ if we don't recognize $v(t)$ as the derivative of some known function?
To do this using a computational approach, we need to know the position at some starting time $t_0$. Let's suppose we know that $x(2)=3$. From there on, we can use the approximation above. Let's try to determine the position at $t=5$.
Let's find the position at a time $h$ later, which we'll call $t_1=t_0+h$. Using the approximation
- $v(t_0)=x'(t_0) \approx \frac{x(t_0+h)-x(t_0)}{h}=\frac{x(t_1)-x(t_0)}{h}$
Solving this equation for \(x(t_1)\) we find that
- \(x(t_1) = x(t_0)+h v(t_0)\).
Replacing all the letters but numbers, we get
- \(x(2.1) = x(2)+0.1 v(2) = 3+0.1 e^{-4}\).
Repeating this process, replacing $t_0$ with $t_1$, we find that \(x(t_2) = x(t_1)+h x(t_1)\) where \(t_2=t_1+h\).
We can repeat this as long as we want, stopping when we get to some desired final time. If we start at $t=2$ and want the position at $t=5$, using $h=0.1$, we'll have to repeat the process times.
Choosing $h$
There is still the issue of choosing a value of $h$. We know that the derivative approximation is better for smaller \(h\) so we can expect that our estimate of $x(5)$ will be better for smaller $h$. However, the smaller we choose it, the more steps it will take to get to $t=5$. To achieve a desired accuracy, start with a convenient (big) value like $h = 0.5$, record the resulting $x(5)$, and repeat the process for $h$ half the size. Repeat. Repeat. After three or four repetitions, you will see less and less change in the successive values of $x(5)$. When the change is less than 0.01%, your accuracy should be sufficient for WeBWorK.
