用积分解释(黎曼和的)极限

Previously, you have learnt about what is Riemann sum and the four main types of Riemann sum as well as how to apply it. Now we are going one step further and draw the link between limits and definite integrals.

Before that, let us revise what you have learnt in Riemann sums.

Well, as some of you may have figured it out, I am basically asking you to approximate the value of $\int_0^{10}x^2\ dx$ for 4 times (ain't nobody got time for that!), and two of them are off by about 7-8% from the actual value of $333\frac{1}{3}$! However, this error can be minimized by increasing the number of rectangles, $N$. As we increase the value of $N$, the width of the rectangle will get narrower and narrower, as shown in the animation below.

Because of this narrowing of the rectangles, can you see why the magnitude of the error is getting smaller and smaller?

To properly start on this section, we shall introduce a new term, Riemann Integrability.

Riemann Integrability

$f$ is Riemann integrable on $[a,b]$ if $\displaystyle\lim_{\max(\Delta x_k)\to0}\ \ \sum_{k=1}^n f(x_k^*)\Delta x_k$ exists, where $\Delta x_k=x_k-x_{k-1}$ is the width of the rectangle and $x_k^*$ is a value chosen in the range $[x_{k-1},x_k]$. In particular, this limit is defined to be equal to $\int^b_af(x)\ dx$, if it exists.

Note: if $f$ is continuous on $[a,b]$, then $f$ is Riemann integrable there. If $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded there.

One should note that the Riemann Integral does not depend on the choice of the partitions (how you partition the interval) or the choice of the points $x_k^*$ in the interval $[x_{k-1}, x_k].$ But for easiness sake, we partition the interval $[a,b]$ into $n$ equal subintervals, which leads to the following useful identity. $$\lim_{n\to\infty}\frac{b-a}{n}\sum_{^{k=1}}^nf\left(a+\frac{k\left(b-a\right)}{n}\right)=\int_a^bf\left(x\right)\ dx$$

With that in mind, we are ready to get our hands dirty. As an exercise for you, represent the following limits as a definite integral and evaluate the limits. You may assume that the limits exist.

1. $\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum^n_{k=1}\frac{k}{n}$
2. $\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum^n_{k=1}\left(\frac{k}{n}\right)^3$
3. $\displaystyle\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k+n}$
4. $\displaystyle\lim_{n\to\infty}\sum^n_{k=1}\frac{k+n}{k^2+2kn+n^2}$

In conclusion, we make the rectangles as thin as possible to get a more accurate approximation or evaluation of the integrals.