Name of integration technique where product term is near constant over the interval?

Question

Consider

$$\int_{\theta-\epsilon}^{\theta+\epsilon} g(x)f(x) dx$$

where f(x) is near constant on the interval $(\theta-\epsilon, \theta+\epsilon)$, and g(x) is not. It follows:

$$\int_{\theta-\epsilon}^{\theta+\epsilon} g(x)f(x) dx \approx f\left(\theta\right)\int_{\theta-\epsilon}^{\theta+\epsilon} g(x) dx$$

I can't find reference to this technique, is this valid? Is there maybe a better way to express this? Maybe some quadrature rule I've missed?

Answer 1

What does it mean for $f(x)$ to be near constant on the interval? It means that $f'(x) \approx 0$ on the interval. As such, here's a relatively naive improvement using integration by parts.

Let G be an anti-derivative of g (i.e. G' = g). By parts

$$\int g(x)f(x)dx \approx G(x)f(x)|_{\theta-\epsilon}^{\theta+\epsilon}$$

which is a slight improvement on the equation I gave. We can recover the original suggested equation as $f(\theta-\epsilon) \approx f(\theta+\epsilon) \approx f(\theta)$, thus

$$G(x)f(x)|_{\theta-\epsilon}^{\theta+\epsilon}\approx f(\theta)\left(G(x)|_{\theta-\epsilon}^{\theta+\epsilon}\right) = f(\theta) \int_{\theta-\epsilon}^{\theta+\epsilon} g(x)dx $$

which is just the proof for the "Constant Multiple Rule of Integration" and rather boring. I was hoping for a more robust, relatively general method. Please chime in if there are better methods.

Answer 2

It's not generally true, you need to refine what you mean by $\approx$ relative to $\epsilon$ and the derivative of $g$ otherwise pathological choices can be made. E.g. let $\theta = 0$, $f(x) = mx$ for some very small $m$ (i.e. "$m \approx 0$"), let $g(x) = \frac{1}{\epsilon^{100} m}x$, then one integral is $0$ but the other is probably pretty big (depending on $\epsilon$).