Given a function $f:[0,1]\to \mathbb{R}$, answer the following
questions:
(i) If $f^2$ is $R$-integrable, is $f$ $R$-integrable?
(ii) If $f^3$ is $R$-integrable, is $f$ $R$-integrable?
(iii) If $f$ is $R$-integrable, is $f^2$ $R$-integrable?
Let $f\in\operatorname{C}\bigl([a,b],\mathbb{R}\bigr)$ and $g:[a,b]\to \mathbb{R}$ be non-negative and integrable. Prove that there is $x_0\in (a,b)$ such that $$ \int_a^b f(x) g(x) \, dx=f(x_0) \int_a^b g(x) \, dx. $$
For an integrable function $u:[a,b]\to \mathbb{R}$, define $$ \|u\|_2^2:=\int_a^b |u(x)|^2 \, dx. $$ Supposing that $f,g,h:[a,b]\to \mathbb{R}$ are integrable, prove the following triangle inequality $$ \|f-h\|_2\le \|f-g\|_2+\|g-h\|_2. $$
Compute the following limit $$ \lim _{n\to\infty}\Big[\frac{n}{n^2+1}+ \frac{n}{n^2+2^2}+\dots+\frac{n}{n^2+(n-1)^2}\Big]. $$
Let $f:[0,2\pi]\to \mathbb{R}$ be Riemann integrable and define $$ g(t)=\int _0^{2\pi}f(x)\sin(tx)\, dx\, ,\: t\in \mathbb{R}. $$ Prove that $g$ is uniformly continuous and that $\lim _{n\to\infty}g(n)=0$.
Prove that the following improper integral converges $$ \int _1^\infty \frac{\sin x}{\log ^{1/2}x}\, dx. $$
Prove that $f:\mathbb{B}(0,1)\subset \mathbb{R}^n\to \mathbb{R} $ defined by $$ f(x)=\begin{cases} \|x\|^{-\alpha},&x\ne 0,\\ 0,&x=0, \end{cases}$$ is integrable when $\alpha <n$. If you can not do it for general $n$, try to prove it when $n\le 3$.
Let $g:\mathbb{R}^2\to \mathbb{R}^2$ and prove that $$ d g_1\wedge d g_2= \det Dg(x)\, dx_1 \wedge d x_2. $$
Use the divergence theorem to evaluate $$ \int _E \frac{y^2}{\sqrt{x^2+4y^2+4z^2}}\, d\sigma_E, $$ where $E=\big\{(x, y, z)\in \mathbb{R}^3: x^2/2 + y^2 + z^2 = 1\big\}$ and $d\sigma_E$ is the area element on E.
Let $\Gamma\subset \mathbb{R}^2$ be a smooth closed curve bounding the region $\Omega$, i.e. let $\Omega$ be the bounded set with $\Gamma=\partial \Omega$. Prove that $$ \operatorname{Area}(\Omega)=\frac{1}{2}\int _\Gamma (x_1dx_2-x_2dx_1). $$