# Why do we use rectangles rather than trapezia when performing integration?

I've been familiar with integration, its motivation and its methods for a while know, but this question has only just entered my mind:

When we integrate to find the area under a curve, why do we split up the curve into infinitesimally-thin rectangles rather than trapezia?

Surely, trapezia are more accurate to work with.

My motivation for this question is that a popular method for numerical integration, the trapezium rule, as its name suggests, splits the curve into arbitrarily-thin trapezia as opposed to rectangles.

So why is the integral defined as $$\int_{a}^{b}y(x)dx:=\lim_{\delta x \to 0} \sum_{x=a}^{b}y(x)\delta x \tag{sum of areas of rectangles}$$ rather than what appears to be the more-accurate $$\int_{a}^{b}y(x)dx:=\lim_{\delta x \to 0} \sum_{x=a}^{b}\frac{y(x)+y(x+1)}{2} \delta x \tag{sum of areas of trapezia}$$?

Also, as far as I know, the integral is defined as the limit of Riemann sums for which you can take any division of $[a,b]$ into as many subintervals as you want, so long as their longest one limits to $0$, and then take any set of values, one from each interval. So your definition is lacking.