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I cannot assume that the integral exists as this is part of the exercise. I'm only allowed to use the definition of the integral, which is the following:

Let $f$ be defined on $[a,b]$. The function $f$ is Riemann integrable on $[a,b]$ if there exists a number $L$ such that for all $\epsilon > 0$ there is $\delta >0$ such that $$|\sigma-L| < \epsilon$$ if $\sigma$ is any Riemann sum of $f$ over a partition $P$ of $[a,b]$ such that $||P||<\delta$.

I'm not exactly sure how to show that the integral exists using this definition.

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  • $\begingroup$ Are you supposed to know about upper and lower Riemann sums? $\endgroup$
    – Mike
    Commented Nov 18, 2014 at 20:24
  • $\begingroup$ Yes, but the only hint I have is to use the Mean Value Theorem. And we shouldn't need to use upper and lower Riemann sums. $\endgroup$
    – Pubbie
    Commented Nov 18, 2014 at 20:25

3 Answers 3

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Let us write a Riemann sum for a partition of $[a,b]$, $a=x_0<x_1<\cdots<x_n=b$ with $x_j-x_{j-1}<\delta$. Note that we can write, using the Mean Value Theorem, $$ x_j^3-x_{j-1}^3=3d_j^2\,(x_j-x_{j-1}). $$ for some $d_j$ with $x_{j-1}\leq d_j\leq x_j$. So $$ \frac{b^3-a^3}3=\sum_{j=1}^n\frac{x_j^3-x_{j-1}^3}3=\sum_{j=1}^nd_j^2\,(x_j-x_{j-1}) $$ Then, for points $c_1,\ldots,c_n$ with $x_{j-1}<c_j<x_j$, consider the Riemman sum $$ \sum_{j=1}^n c_j^2\,(x_j-x_{j-1}). $$ Note that $c_j,d_j\in[x_{j-1},x_j]$, so $|d_j^2-c_j^2|\leq x_j^2-x_{j-1}^2$.

We have \begin{align} \left|\frac{b^3-a^3}3-\sum_{j=1}^n c_j^2\,(x_j-x_{j-1})\right| &=\left|\sum_{j=1}^n (d_j^2-c_j^2)\,(x_j-x_{j-1})\right| \leq\sum_{j=1}^n |d_j^2-c_j^2|\,(x_j-x_{j-1})\\ &\leq\,\delta\,\sum_{j=1}^n|d_j^2-c_j^2|\leq\delta\,\sum_{j=1}^n(x_j^2-x_{j-1}^2)\\ &=\delta\,(x_n^2-x_0^2)=\delta\,(b^2-a^2). \end{align}

That is, given $\varepsilon>0$, a choice of $\delta=\varepsilon/(b^2-a^2)$ will make $(b^2-a^3)/3$ satisfy the definition.

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  • $\begingroup$ Thank you. The first quarter of your proof was what I needed to get started! Thank you so much. $\endgroup$
    – Pubbie
    Commented Nov 18, 2014 at 20:46
  • $\begingroup$ You are very welcome. Glad I could help. $\endgroup$ Commented Nov 18, 2014 at 20:49
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You asked for a proof using the mean-value theorem. Here is one.

Let $f(x) = x^2$, and let $F(x) = \frac{1}{3}x^3$. Let a Riemann sum $\sigma$ be given, corresponding to a partition $a = t_0 < t_1 < \dots < t_n = b$, and sample points $x_1, \dots x_n$ with $x_i \in [t_{i-1},t_i]$.

Define a function $G(x)$ on $[a,b]$ as follows. Take $p$ to be the smallest integer such that $t_p \geq x$. Then set $$G(x) = \sum_{i = 1}^{p-1} f(x_i)(t_i - t_{i-1}) + f(x_p)(x - t_{p-1}).$$

Now $G(x)$ is continuous and has a right-hand derivative everywhere, with $G_{+}'(x)$ constant equal to $f(x_i)$ on each interval $[t_{i-1},t_i)$.

The function $f(x)$ is uniformly continuous on $[a,b]$. Let $\delta$ be given so that $|f(x) - f(y)| \leq \epsilon/(b-a)$ whenever $|x - y| \leq \delta$. If the step of the partition is $\leq \delta$, we obtain the inequality $$|G_{+}'(x) - f(x)| \leq \epsilon/(b-a)$$ throughout $[a,b]$. Applying the mean-value theorem (for right-hand derivatives) to the function $G(x) - F(x)$, we find that $$\left|[G(b) - F(b)] - [G(a) - F(a)]\right| \leq \epsilon.$$ It suffices to note that $G(a) = 0$ and $G(b) = \sigma$ to conclude that $$\left|\sigma - \frac{1}{3}(b^3 - a^3) \right| \leq \epsilon,$$ which completes the proof.

Edit: To make this more explicit, The derivative of $f$ never exceeds $2M^2$, where $M = \max(|a|,|b|)$. So you can take $\delta = \epsilon/2(b-a)M^2$.

Also, here is the specific version of the mean value theorem that I am using. If a function $h$ on $[a,b]$ is continuous and has a right-hand derivative everywhere, with $|h_{+}'(x)| \leq K$, then $|h(b) - h(a)| \leq K(b-a)$.

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  • $\begingroup$ This is actually a proof that if $F$ is continuously differentiable, then it satisfies the fundamental theorem of calculus. $\endgroup$
    – Mike
    Commented Nov 18, 2014 at 21:11
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I think it is a better manner to split the answer into two parts : Existence of the integral, and computation. For the existence part, since $x^{2}$ is continuous on the compact interval (a,b) so it is uniformly continuous and it suffices to follows the general proof which is presented in many text-book (in which the authors prove that any continuous function is Riemann integrable on $[a,b]$.) Well, if we assume the integral exists, then for the computation it suffices to consider a particular sequence of Riemann sums, which necessarily converge to a limit which is the required value of the integrable.

In a first step, assume that $a=0.$ Divide the interval $[0,b]$ into n equal subintervals of length $\Delta _{n}=\frac{b}{n}.$ Consider the following Riemann sum (right end-point of each subinterval) \begin{eqnarray*} \sum_{k=1}^{n}f(\text{right endpoint}_{k})\Delta _{n} &=&\sum_{k=1}^{n}(% \text{right endpoint}_{k})^{2}\Delta _{n} \\ &=&\sum_{k=1}^{n}(k\Delta _{k})^{2}\Delta _{n}=\Delta _{n}\sum_{k=1}^{n}(k\Delta _{k})^{2} \\ &=&\Delta _{n}(\Delta _{n}^{2}+2^{2}\Delta _{n}^{2}+3^{2}\Delta _{n}^{2}+\cdots +n^{2}\Delta _{n}^{2}) \\ &=&\Delta _{n}^{3}(1^{2}+2^{2}+\cdots +n^{2}) \\ &=&\frac{b^{3}}{n^{3}}(1^{2}+2^{2}+\cdots +n^{2}) \\ &=&\frac{b^{3}}{n^{3}}(\frac{n(n+1)(2n+1)}{6}) \\ &=&\frac{b^{3}}{3}\frac{(n+1)(2n+1)}{2n^{2}} \end{eqnarray*} then $$ \int_{0}^{b}x^{2}dx=\lim_{n\rightarrow \infty }\frac{b^{3}}{3}\frac{ (n+1)(2n+1)}{2n^{2}}=\frac{b^{3}}{3}. $$ If a$\neq 0,$ it suffices to make use of the additivity property of the integral \begin{eqnarray*} \int_{a}^{b}x^{2}dx &=&\int_{a}^{0}x^{2}dx+\int_{0}^{b}x^{2}dx \\ &=&-\int_{0}^{a}x^{2}dx+\int_{0}^{b}x^{2}dx \\ &=&-\frac{a^{3}}{3}+\frac{b^{3}}{3} \\ &=&\frac{b^{3}-a^{3}}{3}. \end{eqnarray*}

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