সতর্কীকরণ: আসন্ন মান কতখানি নির্ভরযোগ্য?

When dealing with approximations, most approximations are pretty bad unless you perform the approximation a very large number of times.

Riemann sums are typically the worst type of approximation because you need a VERY large number of rectangles to get a close approximation.

An example of this is for the integral $$\int_0^1\frac{1}{x}\,\text{d}x$$ This integral is infinite, but right rectangular Riemann sums give a finite answer, no matter how many (a finite number) intervals one uses. For example, one interval gives the area 1, and two give the area 3/2. (The limit of the series, which is the harmonic series, is infinite, but no finite subdivision can approximate that.)

The trapezoid rule yields a much better approximation than Riemann sums. With a small number of trapezoids, the approximation really starts to snug up to the curve.

$$\int_a^bf(x)\,\text{d}x\approx\frac{\Delta x}2\left(f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+\dots\\\dots+2f(x_{n-1})+f(x_n)\right)$$

Another approximation is Simpson's Rule, which comes from a parabolic approximation:

$$\int_a^bf(x)\,\text{d}x\approx\frac{\Delta x}3\left(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\dots\\\dots+4f(x_{n-1})+f(x_n)\right)$$

where the coefficients alternate in $4$ and $2$, and the coefficient of $f(x_0)$ and $f(x_n)$ are both $1$. Notice that this only works for an even $n$.