Articles of integration

The method of proving the equality of integrals by showing they agree within $\epsilon$, for an arbitrary $\epsilon>0$

Let $f,g$ be two integrable functions. To show that $\int_{a}^{b} f+g=\int_{a}^{b} f+\int_{a}^{b} g$ why do we need that for any $\epsilon>0$, $$-\epsilon+\int_{a}^{b} f+\int_{a}^{b} g<\int_{a}^{b}f+g<\int_{a}^{b} f+\int_{a}^{b} g+\epsilon\quad $$ ? Intuitively this means that their difference “goes” to $0$, but what does that mean? Is there a theorem which says two integrals $A$ and $B$ are equal […]

Show $\int_0^A \int_0^A \frac{1+(s-t)^2}{1+a^2(s-t)^2} e^{i (t-s) } dt ds\ge 0$

How to show that \begin{align} \int_0^A \int_0^A \frac{1+(s-t)^2}{1+a^2(s-t)^2} e^{i (t-s) } dt ds \ge 0 \end{align} for $a>1$ for all $A>0$. Note the the above integral is real since the function \begin{align} f(t-s)= \frac{1+(s-t)^2}{1+a^2(s-t)^2} \end{align} is symmetric. So, in fact we have to show that \begin{align} \int_0^A \int_0^A \frac{1+(s-t)^2}{1+a^2(s-t)^2} cos(t-s) dt ds \ge 0 \end{align}

How wrong is it? – A “proof” of the FTC that I came up with in high school by hand-waving.

In high school calculus, I was first taught that the area under a curve $f(x)$ between $x=a$ and $x=b$ is given by: $$ A = \lim_{\delta x \rightarrow 0} \sum \limits_{a}^{b} f(x) \delta x $$ Then this limit was defined as being the definite integral of $f(x)$ from $a$ to $b$, and I was told […]

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in L^q(\mathbb{R}^n)$, $\int f_kgd\mu\to\int fgd\mu$, where $1/p+1/q=1$. How to solve this? Thanks. I have tried but failed: Let $\epsilon>0$. Since $||g||_q<\infty$, there exists a compact subset $K$ […]

Determine the volume of the solid obtained by rotating the region bounded between $y = x^2$ and $y = x$ about $y = x.$

I wrote this problem a while ago and solved it using a geometric argument, instead of using coordinate transformation. Here is how I approached this problem: Notice that the radius of a circular segment has radius, $ r = \frac{1}{\sqrt2}(x – x^2)$. So the volume is given as: $V = \int_0^1\left[\frac{1}{\sqrt2}(x – x^2)\right]^2 \pi dx […]

Find symbolic integral for expression containing multivariate polynomials

I’m looking for a way to symbolically integrate the following expression (the $k_i$ are integer constants): Using a generalized form of the identity I was able to express it without the cumbersome area condition of the original expression: I’ve tried expanding the expression using the multinomial theorem, in order to eventually pull the sum outside […]

How to approximate an integral using the Composite Trapezoid Rule

I’m trying to estimate the value of the following integral on the interval $[0,1]$ $$ I = \int_0^1 \frac{1}{1+x} dx $$ So, using the composite trapezoid rule (and with $n=4$, ie I’m only using the first 4 $x_i$ to do the approximation), I get the following expression: $$ I = \frac{67}{60} – \frac{1}{96} (2(1+\xi_1)^{-3} + […]

Evaluate integral using Fourier analysis

$\int_0^\infty \frac{\cos (x)}{1+4x^2}\, dx$ $\int_0^\infty \frac{1}{(1+x^2)^2}\, dx$ There is no hint for these two questions. I think for Q2, since it’s a square, I can use Plancherel formula for $e^{-2\pi|x|}$. But I am not sure how to solve the first one.

If a Riemann integrable function is zero on a dense set, then its integral is zero

Let $g:[a,b]\to\mathbb{R}$ be a Riemann-integrable function such that $g(x)=0$ for all $x\in A\subseteq[a,b]$ where $A$ is dense set. Then $$\int_{a}^{b} g=0$$ How can I show this?

Meaning of algebraic decay

I am reading the paper here and I am running into a few roadblocks. One of them was resolved here and now I am stuck at another. (Pg 177) Suppose $\alpha\in(0,2)$ and $t_i=ih$ for some fixed $h>0$ and every $i$. It can be shown that $\frac{1}{\Gamma(\alpha)}\int_0^{t_n}((t_{n+1}-\tau)^{\alpha-1})-(t_n-\tau)^{\alpha-1})f(\tau,x(\tau))d\tau\\ =\frac{h}{\Gamma(\alpha)(\alpha-1)}\sum_{i=0}^{t_{n-1}}\int_{t_i}^{t_{i+1}}(z_{n-i}^*(\tau))^{\alpha-2}f(\tau,x(\tau))d\tau $ where $z_j^*(\tau)\in (t_{j-1},t_{j+1})$ for all $j$. This […]