The trapezoidal rule

First improvement: instead of a flat-topped rectangle, we join the two values of $f$ at the endpoints of each piece with a segment. We then approximate the area by trapezoids, which follow the curve much better since the segment follows the local slope.

The area of a trapezoid is $\dfrac{\text{short base} + \text{long base}}{2} \times \text{height}$. When we add up all the trapezoids, the interior values $f(x_1), \ldots, f(x_{n-1})$ are counted twice, since each serves as an edge of two neighboring trapezoids, hence the formula:

$$\int_a^b f(x)\,\mathrm{d}x \approx \frac{\Delta x}{2}\Bigl[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\Bigr]$$

The two endpoint values count once, all the interior values count double: this is the pattern $1, 2, 2, \ldots, 2, 1$.

For the same number of points, trapezoids are much more accurate than rectangles: the error decreases like $\dfrac{1}{n^2}$ instead of $\dfrac{1}{n}$. Dividing the error by $4$ then only takes twice as many points.

Method	Top	Order of the error
rectangles	flat	$\dfrac{1}{n}$
trapezoids	slanted segment	$\dfrac{1}{n^2}$

Remember the principle: replacing the flat top with a slope that follows the curve makes accuracy leap forward. One last idea remains for an even better approximation — curving the top itself.